I'm reviewing the Hamilton-Jacobi equation because I'm working on a research project about Kerr black holes and the geodesics of particles gravitating them (This is not really relevant to the question, but I'm saying it for the sake of context). I was working on the equations of motion and many papers use conserved quantities to integrate the equations and find the geodesics... and they usually use the Hamilton-Jacobi formalism.
In the Hamilton-Jacobi formalism, we find a canonical transformation that transformation that goes from the original hamiltonian $H$ to a new one that is identically zero $\tilde{H}$ such that in the new variables the canonical variables and their momenta are just conserved quantities
\begin{equation} \frac{\partial \tilde{H}}{\partial P}=\dot{Q}=0 \\ \frac{\partial \tilde{H}}{\partial Q}=-\dot{P}=0 .\\ \end{equation}
Then one uses the canonical transformation formalism to find the equation
\begin{equation} H(q,\frac{\partial S}{\partial q},t)+\frac{\partial S}{\partial t}=0 \end{equation}
where $S(q,P,t)$ is the generating function. When one solves the equations one finds the original canonical variables as a function of time and $N$ constants of motion $\alpha_i$ which are usually some combination of the new variables $Q$ and $P$.
My question is: Does this formalism guarantee that for any given system with $N$ degrees of freedom $q_i$ ($i=1,2,\dots N$) that has a hamiltonian that behaves nicely (derivable, etc.) there are always $N$ constants of motion $\alpha_i$?
If that is so, then I don't understand why such enphasis is put on how conserved quantities are a special thing that systems with symmetries have. I thought that ugly systems with no symmetries are impossible to integrate precisely due to the lack of conserved quantities. If you can address how the number of conserved quantities relates to integrability that would be awesome.
