Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. I. General theory and τ-function

A general theory of monodromy preserving deformation is developed for a system of linear ordinary differential equations {dY }/{dx }=A(x)Y , where A( x) is a rational matrix. The non-linear deformation equations are derived and their complete integrability is proved. An explicit formula is found for a 1-form ω, expressed rationally in terms of the coefficients of A( x), that has the property d ω=0 for each solution of the deformation equations. Examples corresponding to the “soliton” and “rational” solutions are discussed.