Solution of linear equations in spaces of harmonic variables

General solutions of linear homogeneous and inhomogeneous differential equations of first order on the spaces $SU(N)/U_1(1)\otimes\dots\otimes U_{N-1}(1)$ parametrized by harmonic variables are obtained explicitly for arbitrary $N$. The simplest applications of the obtained results to the investigation of $N$-extended supersymmetric gauge theory are given.