On the Reduction of the Normal Hankel Problem to Two Particular Cases

The normal Hankel problem (NHP) is to describe complex matrices that are normal and Hankel at the same time. The available results related to the NHP can be combined into two groups. On the one hand, there are several known classes of normal Hankel matrices. On the other hand, the matrix classes that may contain normal Hankel matrices not belonging to the known classes were shown to admit a parametrization by real $2\times2$ matrices with determinant 1. We solve the NHP for the cases where the characteristic matrix $W$ of the given class has: (a) complex conjugate eigenvalues; (b) distinct real eigenvalues. To obtain a complete solution of the NHP, it remains to analyze two situations: (1) $W$ is the Jordan block of order two for the eigenvalue 1; (2) $W$ is the Jordan block of order two for $-1$.