We analyze the existence of a parameterized stationary solution $z(\lambda,z_0)=\big(x(\lambda,z_0), p(\lambda,z_0),\,u(\lambda,z_0)\big)\in D\subseteq\mathbb{ R}^{ 2n+1},\,\lambda\in B(0,a)\subseteq\mathop{ \prod}\limits_{ i=1}^{ m}[-a_i,a_i]$, associated with a nonlinear first order PDE, $H_0(x,p(x),u(x))=\hbox{ constant}\,\,(p(x)=\partial_x u(x))$ relying on (a) first integral $H\in\mathcal{ C}^\infty\big(B(z_0,2\rho)\subseteq\mathbb{ R}^{ 2n+1}\big)$ and the corresponding Lie algebra of characteristic fields is of the finite type; (b) gradient system in a Lie algebra finitely generated over orbits $(f.g.o;z_0)$ starting from $z_0\in D$ and their nonsingular algebraic representation.
Categories: math.AP