Is the following result true?
Suppose $M$ is a smooth manifold, $W\subseteq J^1M\cong M\times T^\ast M$ is an open subset, $F\colon W\to \mathbb R$ is a smooth function, and $(x_0,u_0,p_0)\in W$. Let us assume that there is a smooth section $\sigma\in \Gamma(T^\ast M)$ such that $\sigma(x_0)=p_0$. Then there is a function $u$ defined on an open neighbourhood of $x_0$ such that $$ \begin{cases} F(x,u(x),du(x))=0\\ u(x_0)=u_0\\ du(x_0)=p_0 \end{cases} $$
This result seems a case of Theorem 22.39 when $S=\{x_0\}$. Am I right?
Thanks for any help.
