Summary: A study is made of the discrete spectrum of the Hamiltonian \(H_0\) of a many-particle quantum system \(Z_1\) in a magnetic field directed along the \(z\)-axis and increasing unboundedly at infinity in the \((x,y)\)-plane in spaces of functions with arbitrary permutational symmetry. The situation is considered when the boundary of the essential spectrum of \(H_0\) is determined by the decomposition of the system \(Z_1\) into just two stable subsystems. In this case one obtains a description of the discrete spectrum of \(H_0\) in terms of the spectral properties of two effective one-dimensional two-particle operators without a magnetic field. On this basis, conditions are established for the finiteness and infiniteness of the discrete spectrum of \(H_0\), as well as the spectral asymptotics in the case when the discrete spectrum is infinite. In particular, the results can be applied to Hamiltonians of any atoms, their positive ions and most diatomic molecules.