Classical mechanics: Generating function of lagrangian submanifold

I have a short question regarding the geometrical interpretation of the Hamilton-Jacobi-equation.

One has the geometric version of $H \circ dS = E$ as an lagrangian submanifold $L=im(dS)$, which is transverse to the fasers of $T^*Q$ and lies in the coisotropic submanifold $H^{-1}(E)$. Furthermore the pullback of the canonical one-form $\theta$ to L is exact.

The question is: Is L an embedded submanifold? I think if I have a solution S of the Hamilton-Jacobi-equation, then it generates an embedded submanifold $dS(Q)$. But have I have to use an embedded langrangian submanifold to get an "analytical" solution or is an "normal" submanifold enough?

Thx!

-