Parameterized Stationary Solution for first order PDE

We analyze the existence of a parameterized stationary solution $z(λ,z_0)=\big(x(λ,z_0), p(λ,z_0),\,u(λ,z_0)\big)\in D\subseteq\mathbb{R}^{2n+1},\,λ\in B(0,a)\subseteq\mathop{\prod}\limits_{i=1}^{m}[-a_i,a_i]$, 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 $H\in\mathcal{C}^\infty\big(B(z_0,2ρ)\subseteq\mathbb{R}^{2n+1}\big)$ 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.

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