  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.
