Fully nonlinear first-order PDEs and complete integrals

Let $F:O\subseteq {\mathbb{R}}^n\times \mathbb{R}\times {\mathbb{R}}^n\to \mathbb{R}$. A general first-order equation has the form $F(x,u,Du)=0$; it is fully nonlinear when its dependence on the highest-order variable $p=Du$ is not affine after $(x,u)$ is fixed. A local complete integral on open sets $U,A\subseteq {\mathbb{R}}^n$ is a $C^2$ family $S:U\times A\to \mathbb{R}$ such that $(x,S(x;\alpha ),D_{x}S(x;\alpha ))\in O,F(x,S(x;\alpha ),D_{x}S(x;\alpha ))=0,\det D_{x\alpha }^{2}S(x;\alpha )\mathrm{\ne }0$ for every $(x,\alpha )\in U\times A$. Thus the $n$ parameters enter essentially rather than merely labelling repeated copies of one solution. A stationary envelope is a value $u(x)=S(x;\alpha (x))$ selected by $S_{\alpha }(x;\alpha (x))=0$. This definition asserts neither global representation nor differentiability of an envelope.