On varieties of commuting triples II

The variety C2(A) of pairs of commuting matrices in the centralizer of n×n matrix A is known to be irreducible if A is 2-regular matrix. We prove that the variety C2(A) is irreducible also for 3-regular matrices A and in some 4-regular cases, but on the other hand, it can be reducible if A is 5-regular matrix.