# Dark Type Dynamical Systems: The Integrability Algorithm and Applications +
## 2 Nonlinear Dynamical Systems and the Lax Type Representation
### Generative Function of Conserved Quantities  
Assume the Equation (4) is a Lax type representation for the dynamical system (1). We will show that the function \(\xi(\lambda)\in\sigma(S)\) as a functional on the manifold \(M\) depending on the parameter \(\lambda\in\mathbb{C}\), doesn't depend on the evolution parameter \(t\in\mathbb{R}\) by virtue of the dynamical system (1). Equation (4) is evidently equivalent to the following system of differential equations  
\[\partial\tilde{f}/\partial x=l[u;\lambda]\tilde{f},\ \partial\tilde{f}/ \partial t=p[u;\lambda]\tilde{f} \tag{11}\]  
for some matrix \(p[u;\lambda]\in\mathrm{End}\mathbb{C}^{k}\), where \(\tilde{f}\in C(\mathbb{R};\mathbb{C}^{k})\) is the defined above Bloch eigenfunction of the operator \(\mathcal{L}[u;\lambda]:L_{\infty}(\mathbb{R};\mathbb{C}^{k})\to L_{\infty}( \mathbb{R};\mathbb{C}^{k})\) for a fixed \(u\in M\) and the parameter \(\lambda\in\mathbb{C}\). Now one can formulate the following important lemma. Lemma 2: _The spectrum \(\sigma(\mathcal{L})\subset\mathbb{C}\) of the periodic problem (6) does not depend on the temporal parameter \(t\in\mathbb{R}\)._  
Proof: Having taken into consideration the property (10), one easily finds that  
\[\xi(\lambda)\ \partial\tilde{f}(x;\lambda)/\partial t+\tilde{f}(x;\lambda) \partial\xi(\lambda)/\partial t=\xi(\lambda)\ \partial\tilde{f}(x+2\pi;\lambda)/\partial t, \tag{12}\]  
whence, by virtue of (11), one follows that  
\[\partial\xi(\lambda)/\partial t=0, \tag{13}\]  
that is the parametric functional \(\xi(\lambda)\in\mathcal{D}(M)\) is a meromorphic generator function of conserved quantities of dynamical system (1). To construct analytically these conservation laws it's convenient to make use of the Equation (6). To do this we consider the function  
\[\chi(x;\lambda):=\ \partial\ln\tilde{f}_{1}(x,x_{0};\lambda)/\partial x, \tag{14}\]  
where \(\tilde{f}_{1}\in L_{\infty}(\mathbb{R};\mathbb{C})\) is the first component of the vector-function \(\tilde{f}\in L_{\infty}(\mathbb{R};\mathbb{C}^{k})\), normalized at \(x_{0}\in\mathbb{R}\) by unity: \(\tilde{f}_{1}(x_{0},x_{0};\lambda)=1\) for all \(x_{0}\in\mathbb{R}\). Substituting recurrently components of the function \(\tilde{f}(x,x_{0};\lambda)\in L_{\infty}(\mathbb{R};\mathbb{C}^{k})\) in such a form into the Equation (6), after simple calculations we make sure that the function \(\chi(x;\lambda)\in\mathbb{C}^{k-1}(\mathbb{R}/\{2\pi\mathbb{Z}\};\mathbb{C})\) satisfies an ordinary differential equation of the form  
\[\partial^{k-1}\chi/\partial x^{k-1}=R[u;\chi], \tag{15}\]  
where \(R[u;\cdot]:J^{(k-2)}(\mathbb{R}/\{2\pi\mathbb{Z}\};\mathbb{R}^{m})\to\mathbb{C}\) is a polynomial mapping, parametrically depending on a point \(u\in M\) and complex parameter \(\lambda\in\mathbb{C}\). Remark here that at \(k=2\) the Equation (15) becomes the well-known Riccati equation. Having supposed that the function \(\chi(x;\lambda)\in\mathbb{C}^{k-1}(\mathbb{R}/\{2\pi\mathbb{Z}\};\mathbb{C})\), \(x\in\mathbb{R}\), as \(\lambda\to\{\infty\}\) can be represented by making use of the asymptotic expansion  
\[\chi(x;\lambda)\sim\lambda^{s(\chi)}\sum_{j\in\mathbb{Z}_{+}}\chi_{j}[u] \lambda^{-1} \tag{16}\]  
in parameter \(\lambda\in\mathbb{C}\), where \(s(\chi)\in\mathbb{N}\) is some integer, after substitution (16) into the Equation (15) one ensues the recursion relations for densities \(\chi_{j}[u]\), \(j\in\mathbb{Z}_{+}\), giving rise to their explicit expressions as local functionals on the functional manifold \(M\). In particular, all densities \(\chi_{j}[u]\), \(j\in\mathbb{Z}_{+}\), prove to be smooth and periodic in variable \(x\in\mathbb{R}\) functionals on the manifold \(M\).