Deformations of connections, the Riemann Hilbert problem and $\tau$-functions

e give sufficient conditions for the existence of integrable deformations of a rational linear ODE on the projective line and we show when the related connection form obtains a reduced form. Certain coefficients in this reduced form admit a meromorphic continuation to the whole parameter space, while their poles coincide with the zero-set of a Fredholm determinant $\tau$. These properties are similar to the ones holding for the solutions of the KP-hierarchy and form a generalization of work by Malgrange.