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.