$C$-projective equivalence and quantum integrability of the geodesic flow

J-planar curves are a natural generalization of geodesics from (pseudo-)riemannian manifolds to Kähler manifolds. Two Kähler metrics with the same J-planar curves are called c-projectively equivalent. Along with this property come integrals of the geodesic flow. We have shown that when assigning differential operators to these Integrals these do also commute. We shall also show that certain potentials may be added to the constructed integrals such that the commutativity is preserved. For some cases separation of variables may be achieved up to a certain degree.