  In a recent work [1] we consider the topological expansion for the non-mixed
observables (including the free energy) for the formal Cauchy matrix model. The
only restriction in [1] was the fact that all the branch points have to be
simple. This excludes a very interesting situation not encountered in the
literature before, namely the case in which two branch points merge in such a
way that no cycle is pinched. In this work we focus on this situation and
derive new formulas that apply to non-simple and non-singular branch-points.
