Contour integrals as Ad-invariant functions on the fundamental group

We introduce a general approach to contour integrals. It covers usual Abelian integrals, the higher order Melnikov integrals and the generalized Abelian integrals (see [M. Bobieński, H. Żołądek, Limit cycles for multidimensional vector field. The elliptic case, J. Dyn. Control Syst. 9 (2) (2003) 265–310; M. Bobieński, H. Żołądek, Limit cycles of three dimensional polynomial vector fields, Nonlinearity 18 (1) (2005) 175–209]). We prove that the generating function always satisfies a linear differential equation of finite order. We also introduce a relationship between the generalized Abelian integral and an upper triangular representation of the fundamental group of the complex curve.