On the discrete-spectrum of the given $SO(2)$ symmetry of many-particle systems with the potential field and the homogeneous magnetic field

For the system of $n$ identical particles at the homogeneous magnetic field the discrete spectrum of the Hamiltonian $\mathcal{H}^{\alpha,m}$ on the subspaces of the functions with the permutational symmetry $\alpha$ and rotational ($SO(2)$) symmetry $m$ is studied when $m\to\infty$. It is prooved that if some conditions are satisfied there is only one eigenvalue at the discrete spectrum of the operator $\mathcal{H}^{\alpha,m}$. The asymptotics of this eigenvalue for $m\to\infty$ have been found.