Vector particle with electric quadrupole moment in the Coulomb field, nonrelativistic theory

The problem of vector particle with electric quadrupole moment in external Coulomb field is investigated. After separation of the variables, two systems of $4$ and $6$ equations respectively were derived. The terms due to the electric quadrupole moment are present in both systems. The system of $4$ equations is reduced to a second order equation which contains two irregular points $r=0, \infty$ of the rank $3$ and $2$, and four regular points. There are constructed its Frobenius type solutions, convergence of involved series is studied. The transcendence condition for such solutions gives a physically interpretable formula for the energy levels. The system of $6$ equations turns out to be complicated, in the nonrelativistic approximation it is reduced to two coupled equations of the second order, whence the equation of the fourth order follows. Frobenius solutions of this equation are constructed. Two types of solutions are identified that could correspond to the bound states of a particle.