Superintegrable systems with spin induced by coalgebra symmetry

A method for deriving superintegrable Hamiltonians with a spin orbital interaction is presented. The method is applied to obtain a new superintegrable system in Euclidean space $\mathbb{E}_3$ with the following properties. It describes a rotationally invariant interaction between a particle of spin $\frac{1}{2}$ and one of spin 0. Its Hamiltonian commutes with total angular momentum $\vec{\mathcal{J}}$ and with additional vector integrals of motion $\vec{X}$, $\vec{Y}$ with components that are third order differential operators. The integrals of motion form a polynomial algebra under commutation. The system is exactly solvable (in terms of Laguerre polynomials) and the bound state energy levels are degenerate and described by a Balmer type formula. When the spin orbital potential is switched off the system reduces to a hydrogen atom.