Treffer: A system of total differential equations of two variables and its monodromy group

As a generalization of the hypergeometric system in one variable \(t\) studied by K. Okubo, \((t-B) dZ/dt=AZ\), where \(A\) and \(B\) are constant matrices, the author introduces a completely integrable Pfaffian system in two variables \(x\) and \(y\) of the form \(dZ=\Omega Z\), \(\Omega=B(x) Adx+(A-(\rho_ 1+\rho_ 2)) B(y)dy-Ad(x-y)/(x-y)\), \(B(x)=(x-B)^{-1}\), where \(Z\) is an \(n\) dimensional unknown vector, \(A\) is an \(n\) by \(n\) constant matrix, \(\rho_ 1\) and \(\rho_ 2\) are constants such that \((A- \rho_ 1)\) \((A-\rho_ 2)=0\), \(B\) is an \(n\) by \(n\) constant diagonal matrix. It is shown that Appell's systems for \(F_ 1\) and \(F_ 2\) and the system introduced by N. Takayama can be transformed to systems of the above form. The author constructs a set of fundamental solutions of the above system and then gives explicitly the generators of the monodromy group of the system with respect to the fundamental set of solutions. He also studies an invariant Hermitian matrix.