On normal Hankel matrices of low orders

In the previous work of the authors, the problem of describing complex n × n matrices that are simultaneously normal and Hankel was reduced to a system of n ? 1 real equations with respect to 2n unknowns. These equations are quadratic, and it is not at all clear whether they have real solutions. It is shown here that the systems corresponding to n = 3 and n = 4 are solvable and have infinitely many real solutions.