  We investigate a quantum nonrelativistic system describing the interaction of
two particles with spin 1/2 and spin 0, respectively. We assume that the
Hamiltonian is rotationally invariant and parity conserving and identify all
such systems which allow additional integrals of motion that are second order
matrix polynomials in the momenta. These integrals are assumed to be scalars,
pseudoscalars, vectors or axial vectors. Among the superintegrable systems
obtained, we mention a generalization of the Coulomb potential with scalar
potential $V_0=\frac{\alpha}{r}+\frac{3\hbar^2}{8r^2}$ and spin orbital one
$V_1=\frac{\hbar}{2r^2}$.
