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Consider a symplectic manifold $M$, a Hamiltonian vector field $X$ and afibration $\Pi:M\rightarrow N$. Related to these data we have a generalizedversion of the (time-independent) Hamilton-Jacobi equation: the $\Pi$-HJE for$X$, whose unknown is a section $\sigma:N\rightarrow M$ of $\Pi$. The standardHJE is obtained when the phase space $M$ is a cotangent bundle $T^{*}Q$ (withits 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'scharacteristic function. Coming back to the generalized version, among thesolutions 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 isknown for a given Hamiltonian system, then such a system can be integrated upto quadratures. The purpose of the present paper is to prove that, under mildconditions, an isotropic complete solution exists around almost every point of$M$. Restricted to the standard case, this gives rise to an alternative prooffor the local existence of a "complete family" of Hamilton's characteristicfunctions.

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