On two particular cases of solving the normal Hankel problem

The normal Hankel problem is one of characterizing all the complex matrices that are normal and Hankel at the same time. The matrix classes that can contain normal Hankel matrices admit a parameterization by real $2\times2$ matrices with determinant one. Here, the normal Hankel problem is solved in the case where the characteristic matrix of a given class is an order two Jordan block for the eigenvalue 1 or -1.