Pairs of commuting nilpotent operators with one-dimensional intersection of kernels and matrices commuting with a Weyr matrix

I.M. Gelfand and V.A. Ponomarev (1969) proved that the problem of classifying pairs (A,B) commuting nilpotent operators on a vector space contains an arbitrary t-tuple linear operators. Moreover, it representations quiver finite-dimensional algebra, so is considered as hopeless. If such pair, then KerA?KerB?0. We give simple normal form (Anor,Bnor) matrices if KerA?KerB one-dimensional. do not know whether canonical; i.e., uniquely determined by (A,B). prove its uniqueness only Jordan canonical A direct sum blocks same size field zero characteristic. The matrix Anor Weyr A, Bnor commutes with Anor. In order to describe structure (Anor,Bnor), we explicitly all given matrix.