On Matrices Having $J_m(1)\oplus J_m(1)$ as the Cosquare

If the cosquares of complex matrices $A$ and $B$ are similar and there is a unimodular number in the spectrum of the cosquares, then $A$ and $B$ are not necessarily congruent. Assume that such an eigenvalue $\lambda_0$ is unique. In this case, so far, one could verify the congruence of $A$ and $B$ by using a rational algorithm only in two situations: (1)the eigenvalue $\lambda_0$ is simple or semi-simple; (2)there is only one Jordan block associated with $\lambda_0$ in the Jordan form of the cosquares. We propose a rational algorithm for checking congruence in the case where two Jordan blocks of the same order are associated with $\lambda_0$.