The authors deal with the problem of classifying pairs \((A, B)\) of commuting nilpotent operators on a vector space. Such a problem has its origin in the paper of I. M. Gel’fand and V. A. Ponomarev [Funkts. Anal. Prilozh. 3, No. 4, 81–82 (1969; Zbl 0204.45301)] where they proved that classifying pairs \((M, N)\) of commuting nilpotent matrices under similarity transformations includes the problem of classifying \(t\)-tuples of matrices with any \(t\) under similarity transformations \[ (A_1, \ldots, A_t) \rightarrow (S^{-1} A_1 S, \ldots, S^{-1} A_t S), \qquad S {\mbox{ is nonsingular}}. \] Using Belitskii’s algorithm, the authors reduce \((M, N)\) by similarity transformations to some simple pair \((W_M, B)\), where \(W_M\) is the Weyr canonical form of \(M\). They also show that uniqueness of the pair \((W_M, B)\) can be proved only if the Jordan canonical form of \(M\) is a direct sum of Jordan blocks of the same size and the field \(\mathbb{F}\) is of zero characteristic. Finally in order to describe the structure of the matrix \(B\), the authors describe the form of all matrices that commute with a Weyr matrix.