# Dark Type Dynamical Systems: The Integrability Algorithm and Applications +
## 2 Nonlinear Dynamical Systems and the Lax Type Representation
### Conservation Laws and the Related Involutive Properties  
It is easy to observe that the operator \(\Lambda=\Lambda[u]:T^{*}(M)\to T^{*}(M)\), acting according to the rule (29), is hereditarily recursive, satisfying the following determining differential-functional equation:  
\[\partial\Lambda/\partial t=[\Lambda,K^{\prime*}], \tag{30}\]  
following from the well known Lax differential-functional relationship  
\[\partial\varphi/\partial t+K^{\prime*}\varphi=0 \tag{31}\]  
for the gradient vector \(\varphi:=\mathrm{grad}\,H\in T^{*}(M)\), corresponding to any smooth conservation law of the nonlinear dynamical system (1). Moreover, if the nonlinear dynamical system (1) possesses two compatible [14; 32; 33] Poisson operators \(\vartheta\) and \(\eta:T^{*}(M)\to T(M)\), that is their affine some \(\eta+\lambda\vartheta:T^{*}(M)\to T(M)\) persists to be Poisson for all \(\lambda\in\mathbb{R}\), the recursion operator \(\Lambda:T^{*}(M)\to T^{*}(M)\) can be factorized as  
\[\Lambda=\vartheta^{-1}\eta. \tag{32}\]  
In this connection the operators Poisson operators \(\vartheta\) and \(\eta:T^{*}(M)\to T(M)\) are then Noetherian and the nonlinear dynamical system (1) is a bi-Hamiltonian one, i.e.,  
\[u_{t}=-\vartheta\,\mathrm{grad}\,H_{\vartheta}=-\eta\,\mathrm{grad}\,H_{\eta}, \tag{33}\]where the conservation laws \(H_{\theta}\) and \(H_{\eta}\in\mathcal{D}(M)\) are, in general, linear combinations of a finite number of the conservation laws \(\gamma_{j}\in\mathcal{D}(M)\), \(j\in\mathbb{Z}_{+}\), constructed above:  
\[H_{\theta}=\sum_{j=-q}^{m}c_{j}\gamma_{j+q+r},\;\;H_{\eta}=\sum_{j=-q}^{m}c_{j} \gamma_{j+q}, \tag{34}\]  
where \(r\in\mathbb{N}\) and \(\;c_{j}\in\mathbb{R}\), \(j=\overline{-q,m}\), are some constants. Moreover, all evolution flows  
\[u_{t_{j}}:=-\vartheta\,\text{grad}\;\gamma_{j} \tag{35}\]  
with respect to temporal parameters \(t_{j}\in\mathbb{R},j\in\mathbb{Z}_{j}\), are functionally independent on the functional manifold \(M\). As another important consequence of the factorization (32), we observe that the relationship (27) is equivalent to the following one:  
\[\eta\,\text{grad}\;\gamma(\lambda)=\lambda^{r(l)}\vartheta\,\text{grad}\; \gamma(\lambda) \tag{36}\]  
for all \(\lambda\in\mathbb{C}\). From it one easily ensues that all invariants \(\gamma_{j}\in\mathcal{D}(M)\), \(j\in\mathbb{Z}_{+}\), are commuting to each other with respect to two compatible Poisson brackets on \(\mathcal{D}(M)\):  
\[\{\gamma_{j},\gamma_{k}\}_{\vartheta}=0=\{\gamma_{j},\gamma_{k}\}_{\eta}, \tag{37}\]  
where, by definition,  
\[\{\gamma,\mu\}_{\vartheta}:=(\text{grad}\;\gamma|\vartheta\,\text{grad}\;\mu),\;\;\{\gamma,\mu\}_{\eta}:=(\text{grad}\;\gamma|\eta\,\text{grad}\;\mu) \tag{38}\]  
for any smooth functionals \(\gamma,\mu\in\mathcal{D}(M)\). The latter, in particular, means that all evolution flows (35) are functionally independent bi-Hamiltonian and commuting to each other completely Liouville-Arnold type integrable flows on the functional manifold \(M\). The obtained above result can be reformulated as the following proposition.  
Proposition 3: _The Lax type integrable dynamical system (1) is a bi-Hamiltonian flow with respect to two algebraically independent Poissonian structures \(\vartheta\) and \(\eta:T^{*}(M)\to T(M)\) on \(M\) and generates an infinite hierarchy of functionally independent commuting to each other completely Liouville-Arnold type integrable bi-Hamiltonian flows._