case where the perturbation has fixed rank has been solved in [@Silva88_1] over algebraically closed fields.

For regular pencils the problem has been studied for $r=1$ in [@GeTr17]. For arbitrary perturbations of bounded rank the problem has been solved in [@BaRo18], and for perturbations of fixed rank in [@BaRo19]. In both cases the solutions obtained do not involve any condition on the rank of the type “low-rank”, and the results hold for fields having sufficient number of elements (fields requiring just the condition that at least one element of the field or the point at infinity is not included neither in the spectrum of the original pencil nor in the perturbed one).

There is less literature dealing with the case of singular pencils. The problem is more difficult, since the row and column minimal indices of the pencils are involved. For non full rank pencils the change of the four types of invariants under generic low rank perturbations has been characterized in [@TeDo07]. For square singular pencils, in [@LeMaPhTrWi18] the authors represent pencils via linear relations and obtain bounds for the number of Jordan chains which may change under an arbitrary rank-one perturbation. The problem of characterizing the feedback equivalence 