An inverse integrating factor \(V\) (i.i.f. for short), associated to a \(C^1\) two-dimensional ordinary differential equation \(\dot x=P(x,y)\), \(\dot y=Q(x,y),\) is a \(C^1\) solution to the partial differential equation \(P {{\partial V}\over{\partial x}} +Q{ {\partial V}\over{\partial y}}= V \operatorname {div}(P,Q).\) Notice that if \(V\) and the differential equation are defined in a open set \(U\) and \(Z(V)=\{(x,y)\in U: V(x,y)=0\}\) then \(1/V\) is an integrating factor of the vector field \((P,Q)\) defined in \(U\setminus Z(V).\) In [the second author, \textit{J. Llibre} and \textit{M. Viano}, Nonlinearity 9, 501-516 (1996; Zbl 0886.58087)], it is proved that if \(\Gamma\subset U\) is a limit cycle of the considered differential equation and \(V\) is an i.i.f. for it defined in \(U\) then \(\Gamma\subset Z(V).\) The aim of the reviewed paper is to study whether an i.i.f. vanishes or not on other types of special trajectories of the differential equation, like critical points or separatrix cycles. For instance, it is proved that the local dynamics near a critical point \(p,\) which is an isolated zero of an i.i.f. \(V,\) depends directly on the behavior of \(\operatorname {div}(P,Q)\) in a neighborhood of \(p.\) Also the cases of critical points which are not isolated zeros of \(V\) or which are not in \(Z(V)\) are studied. Finally, some results about separatrix cycles are obtained. One of them is: Let \(V\) be an i.i.f. defined on a region containing a separatrix cycle \(\Gamma\) whose critical points are non degenerated saddles then \(\Gamma\subset Z(V).\)