# Dark Type Dynamical Systems: The Integrability Algorithm and Applications +
## 2 Nonlinear Dynamical Systems and the Lax Type Representation
### Gradient-Holonomic Integrability Analysis  
Consider the Equation (6) on the fundamental solution \(F:=F(x,x_{0};\lambda)\in\text{End }\mathbb{C}^{k}\):  
\[\partial F/\partial x=l[u;\lambda]F, \tag{18}\]  
where \(F(x_{0},x_{0};\lambda)=I\) for all \(x_{0}\in\mathbb{R}\) and \(\lambda\in\mathbb{C}\). Since the fundamental solution \(F(x,x_{0};\lambda)\in\text{End }\mathbb{C}^{k}\) is also a functional on the linear manifold \(M\), then from (18) one can easily obtain an equation for the variation \(\delta F\in\text{End}\mathbb{C}^{k}\) of the fundamental solution, when the value \(u\in M\) changes to \((u+\delta u)\in M\). Hence, we have  
\[\partial(\delta Y)/\partial x=l[u;\lambda]\delta Y+\delta l[u;\lambda]\ F \tag{19}\]  
under the evident condition \(\delta F(x_{0},x_{0};\lambda)=0\) for all \(x_{0}\in\mathbb{R}\). The solution of Equation (19) is  
\[\delta F(x,x_{0};\lambda)=\int\limits_{x_{0}}^{x}F(x,y;\lambda)\delta l[u(y); \lambda]F(y,x_{0};\lambda)dy \tag{20}\]  
for all \(x\in\mathbb{R}\). Taking now into consideration that \(S(x_{0};\lambda)=F(x_{0}+2\pi,x_{0};\lambda)\), from relation (20) we find (see [14; 27]) that  
\[\delta S(x_{0};\lambda)=\int\limits_{x_{0}}^{x_{0}+2\pi}F(x_{0}+2\pi,x;\lambda )\delta l[u(x);\lambda]F(x,x_{0};\lambda)dx. \tag{21}\]  
Consider now the functional  
\[\gamma(\lambda):=\text{ tr}S(x_{0};\lambda),\]  
which by virtue of (13) is a generating functional of conservation laws for the dynamical system (1). Having calculated the trace \(\text{tr}S(x_{0};\lambda)\), from (21) we obtain that  
\[\delta\gamma(\lambda)=\int\limits_{x_{0}}^{x_{0}+2\pi}\text{tr}[F(x_{0}+2\pi,x ;\lambda)\delta l[u(x);\lambda]F(x,x_{0};\lambda)]dx, \tag{22}\]  
whence, making use of the Formula (22), we obtain for the gradient vector \(\text{grad }\gamma(\lambda)\in T^{*}(M)\) the following expression:  
\[\text{grad }\gamma(\lambda)=\text{tr}\big{(}l^{\prime,*}S\big{)}, \tag{23}\]  
or  
\[\text{grad }\gamma(\lambda)=\text{tr}(S\partial l/\partial u), \tag{24}\]  
provided that the matrix \(l[u;\lambda]\in\text{End }\mathbb{C}^{k}\) doesn't depend in explicit way on the derivatives of the function \(u\in M\). If the matrix \(l[u;\lambda]\in\text{End }\mathbb{C}^{k}\) depends on derivatives \((u_{x},u_{xx},\ldots u_{sx})\in J^{s}(\mathbb{R}/\{2\pi\mathbb{Z}\};\mathbb{R }^{m}),s\in\mathbb{Z}_{+}\), then Formula (22) will lead to a bit complicated than (24) expression for the gradient vector \(\text{grad }\gamma(\lambda)\in T^{*}(M)\), where \(T^{*}(M)\) denotes the cotangent space to the functional manifold \(M\), being adjoint to the tangent space \(T(M)\) to the manifold \(M\) with respect to the standard bilinear form \((\cdot|\cdot):T^{*}(M)\times T(M)\to\mathbb{R}\). To obtain this gradient vector \(\text{grad }\gamma(\lambda)\in T^{*}(M)\), it is necessary to calculate the expression (23) separately, making use of the Equation (18) and the expression (23). For example, if \(l[u;\lambda]=l(u,u_{x};\lambda)\in\text{End }L_{\infty}(\mathbb{R};\mathbb{C}^{k})\), from the representation (23) we easily obtain that  
\[\text{grad }\gamma(\lambda)=\text{ tr}(S[l,l_{u_{x}}]-S\partial l_{u_{x}}/ \partial x+Sl_{u}), \tag{25}\]  
where \(l_{u}\) and \(l_{u_{x}}\) are the usual partial derivatives with respect to the variables \((u,u_{x})\in J^{1}(\mathbb{R}/\{2\pi\mathbb{Z}\};\mathbb{R}^{m})\) of the matrix function \(l(u,u_{x};\lambda)\in\text{End }L_{\infty}(\mathbb{R};\mathbb{C}^{k})\). Obviously, in general case there exists such a matrix valued vector \(\mathrm{a}=\{\mathrm{a}_{ij}[u;\lambda]:i,j=\overline{1,k}\}\in\!\left(\mathrm{End} \mathbb{C}^{k}\right)^{m}\), being a local functional on \(M\), that  
\[\mathrm{grad}\,\gamma(\lambda)=\;\mathrm{tr}(S\mathrm{a}[u;\lambda]). \tag{26}\]  
Suppose that the vector-matrix \(\mathrm{a}\in\!\left(\mathrm{End}\mathbb{C}^{k}\right)^{m}\) in (26) is non-trivial.