The study tested the spectrum of eigenvalues of quantum systems, in the classical limit admitting the existence of quadratic ones in the momenta of the first integrals. For example, we consider the problem of two-dimensional potential well of finite depth and quantum rotator. We made a comparison of bifurcations in classical and quantum problems. The research showed that the presence of the additional integral imposes a partial ban on the existence of separatrix contours. We examined an algebraic structure of classical integrals, which determines the possibility of bringing Hamiltonian function to Liouville type and separation of variables in Hamilton - Jacobi equation, which entails the separation of variables in Schrödinger equation.