On Normal Hankel Matrices of Low Orders

In the previous work of the authors, the problem of describing complex $n\times n$ 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.