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The topology of Liouville sets of the real forms of the complex genericNeumann system depends indirectly on the roots of the special polynomial$U_{\cal S}(\lambda)$. For certain polynomials, the existence and positions ofthe real roots, according to the suitable parameters of the system, is notobvious. In the paper, a novel method for checking the existence and positionsof the real roots of the polynomials $U_{\cal S}(\lambda)$ is given. The methodand algorithm are based on searching of a positive solution of a system oflinear equations. We provide a complete solution to the problem of existence ofreal roots for all special polynomials in case $n=2$. This is a step closer todetermining the topology of the Liouville sets.

Real Forms of the Complex Neumann System: Real Roots of Polynomial $U_{\cal S}(\lambda)$