Existence of isotropic complete solutions of the Pi-Hamilton-Jacobi equation

Consider a symplectic manifold $M$, a Hamiltonian vector field $X$ and a fibration $\Pi:M\rightarrow N$. Related to these data we have a generalized version of the (time-independent) Hamilton-Jacobi equation: the $\Pi$-HJE for $X$, whose unknown is a section $\sigma:N\rightarrow M$ of $\Pi$. The standard HJE is obtained when the phase space $M$ is a cotangent bundle $T^{*}Q$ (with its canonical symplectic form), $\Pi$ is the canonical projection $\pi_{Q}:T^{*}Q\rightarrow Q$ and the unknown is a closed $1$-form $\mathsf{d}W:Q\rightarrow T^{*}Q$. The function $W$ is called Hamilton's characteristic function. Coming back to the generalized version, among the solutions of the $\Pi$-HJE, a central role is played by the so-called "isotropic complete solutions". This is because, if a solution of this kind is known for a given Hamiltonian system, then such a system can be integrated up to quadratures. The purpose of the present paper is to prove that, under mild conditions, an isotropic complete solution exists around almost every point of $M$. Restricted to the standard case, this gives rise to an alternative proof for the local existence of a "complete family" of Hamilton's characteristic functions.