On generic linear equations with skew-symmetric coefficient matrices

In the first part of the paper we study smooth solvability properties of linear equations. We prove an extension of Mather’s theorem (1973) to skew-symmetric smooth function matrices. For the proof of the skew-symmetric case as well as of Mather’s original results for the cases where the coefficient matrices are general matrices or symmetric matrices, the algebraic methods of Bochnak (1973) are applied. In the second part using criteria for solutions of linear equations we obtain sufficient conditions for smooth solvability of generalized Hamiltonian systems on smooth constraints.