I am reading the book Classical dynamics: a contemporary approach by J. V. José and E. J. Saletan. In chapter 6.1, after deducing Hamilton-Jacobi (HJ) equation, the following is stated:
This (obtaining the solution of the HJ equation) can be done only if the Hessian satisfies the condition $|\partial^2S/\partial q^\alpha\partial Q^\beta|\neq 0$, but recall that this is a condition already imposed on a Type 1 generator.
A Type 1 generating function is $F_1(q,Q,t)=S(q,Q,t)$. I don't understand why such condition is already imposed in this kind of generating function, this only happens to type 1, or it also works in type 2 and others? Also, why is this condition even needed?
The book doesn't explain any further, so I would like to understand this mathematical needs for solving HJ equation.