You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


---
bibliography:
- 'vcBBM.bib'
---

**Enhanced group classification of\
Benjamin–Bona–Mahony–Burgers equations**

Olena Vaneeva$^{\dag 1}$, Severin Pošta$^{\ddag 2}$ and Christodoulos Sophocleous$^{\S 3}$

*${}^\dag$ Institute of Mathematics of NAS of Ukraine, 3 Tereshchenkivska Str., 01601 Kyiv-4, Ukraine\
${}^\ddag$ Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering,\
$\phantom{{}^\ddag}$ Czech Technical University in Prague, 13 Trojanova Str., 120 00 Prague, Czech Republic\
${}^\S$ Department of Mathematics and Statistics, University of Cyprus, Nicosia CY 1678, Cyprus*

$\phantom{{}^\dag{}\;}\ $E-mails: *$^<PRESIDIO_ANONYMIZED_EMAIL_ADDRESS>, $^<PRESIDIO_ANONYMIZED_EMAIL_ADDRESS>, $^<PRESIDIO_ANONYMIZED_EMAIL_ADDRESS>*

Introduction
============

The regularized long wave equation $u_t+u_x+uu_x-u_{xxt}=0$ was proposed by Peregrine [@peregrine] and later by Benjamin et al. [@Benjamin] to describe small-amplitude long waves on the surface of water in a channel. In order to take into account the mechanisms leading to the degradation of the wave the model including dissipative term $u_{xx}$ was considered in [@Amick1989], namely $$\label{bbmb}
u_t+u_x+uu_x-\nu u_{xx}-u_{xxt}=0, \quad \nu\in\mathbb{R}^+.$$ Here $u = u(x, t)$ is a real-valued function of the two real variables $x$ and $t$, which, in applications, are typically proportional to distance in the direction of propagation and to elapsed time, respectively. The dependent variable may represent a displacement of the underlying medium or a velocity [@Amick1989]. The regularized long wave equation with a Burgers-type dissipative term appended is more frequently called the Benjamin–Bona–Mahony–Burgers (BBMB) equation [@Naumkin].

Last decade model equations with time-dependent coefficients are intensively studied with different points of view. This is due to the fact that they can model real phenomena with more accuracy taking into account parameters changing in time, like, for example, slightly varying depth of water or thickness of ice layer [@kuriksha]. The most general form of the BBMB equation with time-dependent coefficients is $$\label{bbmb_fghk}
u_t+f(t)u_x+g(t)uu_x+k(t)u_{xx}+h(t)u_{xxt}=0,\quad ghk\neq0,$$ where $f$, $g$, $h$, and $k$ are smooth functions of the variable $t$.

In this paper we aim to investigate this class with the Lie symmetry point of view, namely, to present the complete group classification of this class of equations. A similar study was initiated in [@Kumar] but the complete and correct group classification was not achieved therein. Lie symmetries and conservation laws of equations  without dissipative term (i.e. with $k=0$) were thoroughly investigated in [@VPS2015].

There are several reasons why it is important to have the complete and correct group classification of a class of nonlinear PDEs. Firstly, the classification list reveals all equations from the class under study which can be reduced to PDEs with fewer number of independent variables or even to ODEs. Then the powerful and, what is most important, algorithmic Lie reduction method can be used to construct exact solutions [@olve1993b; @Ovsiannikov; @Ibragimov]. Secondly, the equations admitting Lie symmetry extensions potentially are more interesting for applications [@FN]. There are also other advantages of Lie symmetries such as generation of new solutions from known ones, construction of conservation laws, etc. It should be emphasized that though there exist several computer algebra packages for computation of Lie symmetries, and finding of maximal Lie symmetry algebra of a certain fixed differential equation is a routine task which can be done automatically in majority of cases this is not the case for the group classification problem. Though there exist successful examples for some simple classes involving a few number of arbitrary elements, for the rest of cases the involvement of a researcher and recently developed tools of group analysis are needed.

We carry out the group classification of class  using the method of mapping between classes suggested in [@vane2009] and then successfully applied for several classes of variable coefficient PDEs, see, e.g., recent works [@VKS2015; @VPS2015]. When the complete group classification is achieved Lie reductions of BBMB equations to ODEs are performed as well as some exact solutions are constructed. It is understood that not all group-invariant solutions will fit the specific physical situation and respective boundary conditions, this problem, in particular, was discussed recently in [@pucci]. Nevertheless, even “non-physical” exact solutions can be usefull, for example, for testing numerical methods.

Group classification
====================

The statement of the group classification problem is formulated as follows: given a class of differential equations, the goal is to classify all possible cases of extension of Lie invariance algebras of such equations with respect to the equivalence group of the class [@Ovsiannikov].

Consider firstly the transformational properties of class . We look for the admissible point transformations (triples consisting of two equations from the class and a nondegenerate point transformations linking them [@popo2010a]) using the direct method [@kingston1997]. The proofs of the statements below are similar to those presented in [@VPS2015] for equations  with $k=0$. Thus, we skip the details of calculations for the sake of brevity and present the final results only.

All the admissible transformations in class  are generated by equivalence transformations and therefore this class is normalized in the usual sense. The following statement is true.

The usual equivalence group $G^{\sim}$ of class  consists of the transformations $$\begin{gathered}
\tilde t=T(t),\quad \tilde x=\delta_1x+\delta_2,  \quad
\tilde u=\delta_3u+\delta_4, \\
\tilde k(\tilde t)=\dfrac{\delta_1{}^2}{T_t}k(t),\quad
\tilde f(\tilde t)=\dfrac{\delta_1}{T_t\delta_3}(\delta_3 f(t)-\delta_4g(t)), \quad
\tilde g(\tilde t)=\dfrac{\delta_1}{T_t\delta_3}g(t),\quad
\tilde h(\tilde t)=\delta_1{}^2h(t),\end{gathered}$$ where $\delta_j$, $j=1,2,3,4$, are arbitrary constants with $\delta_1\delta_3\not=0$ and $T=T(t)$ is an arbitrary smooth function with $T_t\neq0.$ Class  is normalized in the usual sense.

Using this theorem, we can formulate the criterion of reducibility of variable coefficient BBMB equations from class  to their constant coefficient counterparts:

[*A variable-coefficient equation from class  is reduced to a constant-coefficient equation from the same class by a point transformation if and only if the corresponding coefficients $f,$ $g$, $h$, and $k$ satisfy the conditions $$\label{criterion1}
\left( f/g\right)_t=h_t=k_t=0.$$ i.e., $k$ and $h$ are constants and the function $f$ is proportional to $g$.*]{}

We note that the maximal Lie invariance algebra $A^{\rm max}$ of the constant coefficient BBMB equation was found in [@Bruzon2009]. It is two-dimensional Abelian algebra $\langle\partial_t,\partial_x\rangle$ spanned by the operators of time and space translations.

   no.          $H(t)$                 $K(t)$                     $F(t)$              Basis of $A^{\max}$
  ----- ---------------------- ---------------------- ------------------------------- ------------------------------------------------------------------
    0         $\forall$              $\forall$                   $\forall$            $\partial_x$
    1    $\varepsilon t^\rho$   $\lambda t^{\rho-1}$   $\delta t^{\frac{\rho-4}{2}}$  $\partial_x,\ 2t\partial_t+\rho x\partial_x+(\rho-2)u\partial_u$
    2     $\varepsilon e^t$        $\lambda e^t$           $\delta e^{\frac12t}$      $\partial_x,\  2\partial_t+x\partial_x+u\partial_u$
    3       $\varepsilon$            $\lambda$                   $\delta$             $\partial_x,\ \partial_t$

  : \[TableLieSym1\]The group classification of class  up to $G^\sim_1$-equivalence.

\

The presence of four arbitrary elements in class  leads to difficulties in solving the group classification problem. Therefore, we firstly simplify the problem by reducing the number of arbitrary elements in the class. This can be done either via gauging of arbitrary elements by equivalence transformations or using the method of mapping between classes (see [@VPS2015] for the details). We choose the second option.

The family of point transformations $$\label{trr}
\tilde t=\int\!\! g(t){\rm d}t,\quad \tilde x=x,\quad\tilde u=u+\frac{{f}(t)}{g(t)},$$ parameterized by two arbitrary elements of class , maps class  to the related class of variable coefficient BBMB equations with a forcing term $$\label{eqBBMimaged}
u_t+ uu_x+K(t)u_{xx}+ H(t)u_{xxt}=F(t),\quad HK\neq0.$$ (Tildes in  are omitted.) The arbitrary elements of the initial class  and the imaged class  are related via the formulas $$\label{form_arb_el}
K(\tilde t)=\frac{k(t)}{g(t)},\quad H(\tilde t)=h(t),\quad F(\tilde t)=\frac1{g(t)}\left(\frac{f(t)}{g(t)}\right)_t.$$ Following the method of mapping between classes, we firstly classify Lie symmetries of the imaged class  and then use the family of point transformations  and the relations to extend the result to the initial class .

In order to efficiently solve the group classification problem for class , we look for admissible transformations in this class using the direct method. Similarly to class  such transformations are exhausted by transformations from the usual equivalence group admitted by this class.

The usual equivalence group $G^{\sim}_1$ of class  comprises the transformations $$\begin{gathered}
\tilde t=\dfrac{\delta_1}{\delta_3}t+\delta_0,\quad \tilde x=\delta_1x+\delta_2,  \quad
\tilde u=\delta_3u,\\
\tilde K(\tilde t)=\delta_1\delta_3 K(t),\quad
\tilde H(\tilde t)=\delta_1{}^2H(t),\quad
\tilde F(\tilde t)=\dfrac{\delta_3{}^2}{\delta_1}F(t),\end{gathered}$$ where $\delta_j$, $j=0,1,2,3$, are arbitrary constants with $\delta_1\delta_3\not=0$. Class  is normalized in the usual sense.

Further we use Lie infinitesimal invariance criterion and get the determining equations involving coefficients of symmetry generators $\Gamma$, arbitrary elements of the class and derivatives of both entities. Integration of those determining equations that do not involve arbitrary elements leads

   no.          $h(t)$                 $k(t)$                      $f(t)$              Basis of $A^{\max}$
  ----- ---------------------- ----------------------- ------------------------------- ------------------------------------------------------------------
    0         $\forall$               $\forall$                   $\forall$            $\partial_x$
    1    $\varepsilon t^\rho$   $ \lambda t^{\rho-1}$   $\delta t^{\frac{\rho-2}{2}}$  $\partial_x,\ 2t\partial_t+\rho x\partial_x+(\rho-2)u\partial_u$
    2     $\varepsilon t^2$          $\lambda t$                $\delta\ln t$          $\partial_x,\ t\partial_t+x\partial_x-\delta\partial_u$
    3     $\varepsilon e^t$         $\lambda e^t$           $\delta e^{\frac12t}$      $\partial_x,\  2\partial_t+x\partial_x+u\partial_u$
    4       $\varepsilon$             $\lambda$                  $\delta t$            $\partial_x,\ \partial_t-\delta\partial_u$

  : \[TableLieSym2\]The group classification of class  up to $G^\sim$-equivalence.

\

[ccccl]{} no.&$h(t)$&$k(t)$&$f(t)$&Basis of $A^{\max}$\
0&$\forall$&$\forall$&$\forall$&$\partial_x$\

------------------------------------------------------------------------

1&$\mu_1(\varepsilon T+\kappa)^\rho$&$\mu_2g(\varepsilon T+\kappa)^{\rho-1}$& $\mu_3g(\varepsilon T+\kappa)^{\frac{\rho-2}2}+\mu_4g$& $\partial_x,\ \dfrac2g(\varepsilon T+\kappa)\partial_t+\varepsilon\rho x\partial_x+\varepsilon(\rho-2)(u+\mu_4)\partial_u$\

------------------------------------------------------------------------

2&$\mu_1(\varepsilon T+\kappa)^2$&$\mu_2g(\varepsilon T+\kappa)$&$\mu_3g\ln(\varepsilon T+\kappa)+\mu_4g$& $\partial_x,\  \dfrac1g(\varepsilon T+\kappa)\partial_t+\varepsilon x\partial_x-\varepsilon\mu_3\partial_u$\

------------------------------------------------------------------------

3&$\mu_1 \exp({\sigma T})$&$\mu_2g \exp({\sigma T})$&$\mu_3 g\exp({\frac12\sigma T})+\mu_4g$& $\partial_x,\  \dfrac2g\partial_t+\sigma x\partial_x+\sigma(u+\mu_4)\partial_u$\

------------------------------------------------------------------------

4&$\mu_1$&$\mu_2g$&$\mu_3gT+\mu_4g$&$\partial_x,\ \dfrac1g\partial_t-\mu_3\partial_u$\

\

Any constant coefficient BBMB equation $u_t+fu_x+guu_x+k u_{xx}+h u_{txx}=0$ can be reduced to the equation $u_t+uu_x-u_{xx}+\varepsilon u_{txx}=0$,\
where $\varepsilon=\mathrm{sign}(h)$, (Case 4 of Table 2 with $\delta=0$ and $\lambda=-1$) by the transformation $\tilde t=-\dfrac{k}{{|h|}}t,$ $\tilde x=\dfrac{x}{\sqrt{|h|}},$ $\tilde u=-\dfrac{\sqrt{|h|}}{k}(gu+f)$ from the group $G^\sim$.

to the general form of the infinitesimal generators $\Gamma=(c_1t+c_0)\partial_t+(c_2x+c_3)\partial_x+(c_2-c_1)u\partial_u,$ where $c_i,$ $i=0,\dots3,$ are arbitrary constants. Then the classifying equations involving arbitrary elements take the form $$(c_1t+c_0)H_t=2c_2H,\quad (c_1t+c_0)K_t=(2c_2-c_1)K,\quad (c_1t+c_0)F_t=(c_2-2c_1)F.$$ We integrate these equations up to the $G^\sim_1$-equivalence and obtain the classification list. The results are summarized in Table \[TableLieSym1\].

The classification list for the initial class  can be obtained using the transformation , the relations and the group classification results derived for the imaged class  (Table 1). The transformation  can be considered as a composition of the equivalence transformation $\mathcal\tau^\sim\colon\ \tilde t=\int\!\! g(\bar t){\rm d}\bar t$, $\tilde x=\bar x$, $\tilde u=\bar u$ from the group $G^\sim$ and the transformation $\tau\colon \ \bar t=t$, $\bar x=x$, $\bar u=u+\frac{{f}(t)}{g(t)}$ that does not belong to $G^\sim$. The transformation $\mathcal\tau^\sim$ maps any equation from (1) to the equation from the same class with $g=1.$ In order to obtain the group classification for class  up to the $G^\sim$-equivalence it is enough to consider the transformation  with $g=1.$ Then the formulas  which connect arbitrary elements in classes  and  take the simple form $H=h$, $K=k$, $F=f_t.$ We integrate the latter ODE for the values of $F$ appearing in Table 1. We, respectively, get the following forms of $f=f(t)$:

1\. $f=\bar\delta t^{\frac{\rho-2}{2}}+C$ ($\bar\delta=2\delta/(\rho-2)$), if $\rho\neq2$ and $f=\delta \ln t+C,$ otherwise;

2\. $f=\bar\delta e^{\frac12t}+C$ ($\bar \delta=2\delta$);

3\. $f=\delta t+C$.

The integration constant $C=0\bmod G^\sim.$ The last step is to perform the change of variable $\tilde u=u+f(t)$ in basis operators of the maximal Lie invariance algebras presented in Table 1. The results are summarized in Table 2.

We also derive the complete list of Lie symmetry extensions for the entire class , where arbitrary elements are not simplified by point transformations (Table 3). The group classification results reveal equations of the form  that may be of more interest for applications and for which the classical Lie reduction method can be used.

Reductions and exact solutions
==============================

The classes of BBMB equations  and  are similar with respect to the transformations . If one have exact solutions for equations  the similar solutions for equations  can easily be recovered using . That is why it is convenient to perform classification of Lie reductions for class , where the number of inequivalent cases of Lie symmetry extension is smaller.

To perform the classification of Lie reductions we need the optimal systems of one-dimensional subalgebras of the maximal Lie invariance algebras of the BBMB equations . Such algebras are at most two-dimensional. The optimal system of a two-dimensional Lie algebra $\langle X_1, X_2\rangle$ is $\{\langle X_1\rangle, \langle X_2\rangle\}$ if the algebra is non-Abelian and $\{\langle X_1\rangle, \langle X_2+\alpha X_1\rangle\}$, where $\alpha\in\mathbb R$, if it is Abelian. We have non-Abelian algebras in Case 1 with $\rho\neq0$ and Case 2 of Table 1 and Abelian algebras in Case 1 with $\rho=0$ and Case 3 of this table. We note that reductions with respect to subalgebra $\langle X_1=\partial_x\rangle$ lead to trivial constant solutions. Therefore, we perform only the reductions with the second subalgebra. For each case we list the BBMB equation, the one-dimensional subalgebra, the Ansatz constructed with this subalgebra and the corresponding reduced ODE.

Case $1_{\rho\neq0}$. $$\begin{gathered}
\label{eqBBM1}
u_t+ uu_x+\lambda t^{\rho-1} u_{xx}+\varepsilon t^\rho u_{xxt}=\delta t^{\frac{\rho-4}{2}},\\\nonumber
\langle 2t\partial_t+\rho x\partial_x+(\rho-2)u\partial_u\rangle,\quad u=t^{\frac\rho2-1}\varphi(\omega),\ \mbox{where}\ \omega=xt^{-\frac\rho2},\\\nonumber
\varepsilon\rho\,\omega\varphi'''+(\varepsilon(\rho+2)-2\lambda)\varphi''-2\varphi\varphi'+\rho\,\omega\varphi'+(2-\rho)\varphi+2\delta=0.\end{gathered}$$

This ODE has particular exact solution $\varphi=\omega+\frac{2\delta}\rho,$ which gives the “degenerate” solution $u=\dfrac xt+\dfrac{2\delta}\rho t^{\frac\rho2-1}$ of equation  with $\rho\neq0.$

Case $1_{\rho=0}$. $$\begin{gathered}
\label{eqBBM1a}
u_t+ uu_x+\lambda t^{-1} u_{xx}+\varepsilon u_{xxt}=\delta t^{-2},\\\nonumber
\langle t\partial_t+\alpha\partial_x-u\partial_u\rangle,\quad u=\frac1t\varphi(\omega),\ \mbox{where}\ \omega=x-\alpha\ln t,\\\nonumber
\varepsilon\alpha \varphi'''+(\varepsilon-\lambda)\varphi''-\varphi\varphi'+\alpha\varphi'+\varphi+\delta=0.\end{gathered}$$ For $\alpha=-\delta$ this ODE has particular solution $\varphi=\omega+c$, where $c$ is an arbitrary constant. This leads to the “degenerate” solution $u=\dfrac1t(x+\delta\ln t+c)$ of the equation . Case 2. $$\begin{gathered}
\label{eqBBM3}
u_t+ uu_x+\lambda e^t u_{xx}+\varepsilon e^t  u_{xxt}=\delta e^{\frac12t},\\\nonumber
\langle 2\partial_t+x\partial_x+u\partial_u\rangle,\quad u=e^{\frac t2}\varphi(\omega),\ \mbox{where}\ \omega=xe^{-\frac t2},\\\nonumber
\varepsilon\omega\varphi'''+(\varepsilon-2\lambda)\varphi''-2\varphi\varphi'+\omega\varphi'-\varphi+2\delta=0.\end{gathered}$$

Case 3. $$\begin{gathered}
\label{eqBBM4}
u_t+ uu_x+\lambda  u_{xx}+\varepsilon  u_{xxt}=\delta,\\\nonumber
\langle \partial_t+\alpha\partial_x\rangle,\quad u=\varphi(\omega),\ \mbox{where}\ \omega=x-\alpha t,\\\nonumber
\varepsilon\alpha\varphi'''-\lambda\varphi''-\varphi\varphi'+\alpha\varphi'+\delta=0.\end{gathered}$$ Up to the equivalence we can consider $\lambda=-1$. We found exact solutions for the case $\delta=0$, these are $\varphi=-2\tanh\omega$, $\alpha=0$; and $$\varphi=\pm\frac{1-12\varepsilon}{10\varepsilon}-\frac{12}5\tanh\omega\pm\frac65\tanh^2\omega,\quad \alpha=\pm\frac1{10\varepsilon}.$$ Thus, equation $u_t+ uu_x-u_{xx}+\varepsilon  u_{xxt}=0$ admits the exact solutions $u=-2\tanh x$ and $$u=\pm\frac{1-12\varepsilon}{10\varepsilon}-\frac{12}5\tanh\left(x\pm\frac t{10\varepsilon}\right)\pm\frac65\tanh^2\left(x\pm\frac t{10\varepsilon}\right).$$

Conclusion and discussion
=========================

Using the method of mapping between classes we have presented the complete group classification of BBMB equations . As a by-product of this approach we also got the group classification of a related class of BBMB equations with a forcing term . For the convenience of applications we adduced the results in two ways: the classification list where only inequivalent equations are presented (Table 2) and the list with their most general forms (Table 3). Lie symmetries of the BBMB equations are not very reach. The maximal Lie invariance algebras of such equations are at most two-dimensional. Nevertheless, the derived group classification has revealed equations  that may be of more potential interest for applications and for which the classical Lie reduction method can be used. The classification of reductions was also performed as well as some exact solutions were constructed.

We note that each equation from the class  admits the “natural” conservation law with the constant characteristic $\lambda=1$. The corresponding density and flux are $$F^1=u,\quad G^1=f(t)u+\frac12g(t)u^2+k(t)u_x+h(t)u_{tx}.$$ For general admitted values of the arbitrary elements $f$, $g$, $k$, and $h$ the associated space of conservation laws is one-dimensional. The interesting problem is to classify local conservation laws of equations from class  and to find those values of arbitrary elements for which the dimension of space of conservation laws will be higher. Conservation laws lead to auxiliary (potential) systems which admit Lie symmetries that might induce nonlocal (potential) symmetries of the class . The auxiliary system that corresponds to the above conservation law admits Lie symmetries that project into Lie symmetries of equations  and therefore do not induce nontrivial potential symmetries of these equations.

Acknowledgments {#acknowledgments .unnumbered}
---------------

[ OV would like to thank the Department of Mathematics, FNSPE, Czech Technical University in Prague and the Department of Mathematics and Statistics of the University of Cyprus for the hospitality and support.]{}