On classification of pairs of potent linear operators with the simplest annihilation condition

We study the problem of classifying the pairs of linear operators $\mathcal{A}, \mathcal{B}$ (acting on the same vector space), when the both operators are potent and $\mathcal{A}\mathcal{B}=0$. We describe the finite, tame and wild cases and classify the indecomposable pairs of operators in the first two of them.