Formal solutions of a class of Pfaffian systems in two variables

In this paper, we present an algorithm for computing a fundamental matrix of formal solutions of completely integrable Pfaffian systems with normal crossings in two variables. First, we associate to the Pfaffian system a singular linear system of ordinary differential equations from which its formal invariants can be efficiently derived. After that, we give a generalization of the Moser-based rank reduction algorithm of [5]. These two items allow us to construct formal solutions by following the recursive algorithm given in [4] for singular linear systems of ordinary differential equations. Our algorithm builds upon the package ISOLDE [9] and is implemented in the computer algebra system Maple.