  In this paper, we discuss a system of differential equations for the Fisher
integral on the special orthogonal group. Especially, we explicitly give a set
of linear differential operators which generates the annihilating ideal of the
Fisher integral, and we prove that the annihilating ideal is a maximal left
ideal of the ring of differential operators with polynomial coefficients. Our
proof is given by a discussion concerned with an annihilating ideal of a
Schwartz distribution associated with the Haar measure on the special
orthogonal group. We also give differential operators annihilating the Fisher
integral for the diagonal matrix by a new approach.
