Complete integral of the Hamilton-Jacobi equation

A complete integral depends on as many independent parameters as configuration coordinates and supplies a locally invertible canonical transformation to constants of motion. For a type-one generating function $G(q,Q,t)$, require a nondegenerate mixed-derivative matrix in the relevant local chart. Solving $G_t+H(q,G_q,t)=0$ makes the transformed Hamiltonian zero, so both new canonical coordinates and new canonical momenta are constant. An arbitrary parameter-deficient solution does not ensure such a transformation.