  A systematic search for superintegrable quantum Hamiltonians describing the
interaction between two particles with spin 0 and 1/2, is performed. We
restrict to integrals of motion that are first-order (matrix) polynomials in
the components of linear momentum. Several such systems are found and for one
non-trivial example we show how superintegrability leads to exact solvability:
we obtain exact (nonperturbative) bound state energy formulas and exact
expressions for the wave functions in terms of products of Laguerre and Jacobi
polynomials.
