Pseudo-Abelian integrals along Darboux cycles: A codimension one case

We investigate a polynomial perturbation of an integrable, non-Hamiltonian system with first integral of Darboux type. In the paper [M. Bobieński, P. Mardešić, Pseudo-Abelian integrals along Darboux cycles, Proc. Lond. Math. Soc., in press] the generic case was studied. In the present paper we study a degenerate, codimension one case. We consider 1-parameter unfolding of a non-generic case. The main result of the paper is an analog of Varchenko–Kchovanskii theorem for pseudo-Abelian integrals.