Finite spin models, applicable to investigations of mesoscopic rings, give rise to eigenproblems of very large dimensions. Efficient, and as exact as possible, solutions of such eigenproblems are very difficult. A method leading to block diagonalization of Hamiltonian matrix is proposed in this paper. For a given symmetry group of a Heisenberg Hamiltonian commuting with the total spin projection (i.e. with the total magnetization being a good quantum number) appropriate combinatorial and group-theoretical structures (partitions, orbits, stabilizers etc.) are introduced and briefly discussed. Generation of these structures can be performed by means of algorithms being modifications of standard ones. Main ideas are presented in this paper, whereas the actual form of algorithms will be discussed elsewhere.