On algebraic-analytic aspects of the abelian Liouville-Arnold integrability by quadratures of Hamiltonian systems on cotangent spaces

A symplectic theory approach is developed for solving the problem of algebraicanalytical construction of integral submanifold imbedding mapping for integrable via the abelian Liouville-Arnold theorem Hamiltonian systems on canonically symplectic phase spaces. The related Picard-Fuchs type equations are derived for the first time straightforwardly, making use of a method based on generalized Francoise-Galissot-Reeb differential-geometric results. The relationships between toruslike compact integral submanifolds of a Liouville-Arnold integrable Hamiltonian system and solutions to corresponding Picard-Fuchs type equations is stated.