Parameterized Stationary Solution for first order PDE

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, $H_0(x,p(x),u(x))=\text{constant}(p(x)={\mathrm{\partial }}_{x}u(x))$ relying on (a) first integral $H\in {\mathcal{C}}^{\mathrm{\infty }}(B(z_0,2\rho )\subseteq {\mathbb{R}}^{2n+1})$ and the corresponding Lie algebra of characteristic fields is of the finite type; (b) gradient system in a Lie algebra finitely generated over orbits $(f\mathrm{.}g\mathrm{.}o;z_0)$ starting from $z_0\in D$ and their nonsingular algebraic representation.