  Pizzetti's formula explicitly shows the equivalence of the rotation invariant
integration over a sphere and the action of rotation invariant differential
operators. We generalize this idea to the integrals over real, complex, and
quaternion Stiefel manifolds in a unifying way. In particular we propose a new
way to calculate group integrals and try to uncover some algebraic structures
which manifest themselves for some well-known cases like the Harish-Chandra
integral. We apply a particular case of our formula to an Itzykson-Zuber
integral for the coset SO(4)/[SO(2)xSO(2)]. This integral naturally appears in
the calculation of the two-point correlation function in the transition of the
statistics of the Poisson ensemble and the Gaussian orthogonal ensemble in
random matrix theory.
