This paper presents study of stability of a special two-dimensional system with switching. Its subsystems have second order and are Hurwitz. The first subsystem is linear system presented in Frobenius form and the second one has the same eigenvalues as first one but has swapped states and corresponding realization. The considered problem is to find all permissible eigenvalues that provide asymptotic stability to the switched system under any switching signal and non-zero initial condition. The study is based on common Lyapunov functions which existence is checked using known approach that consists in studying the stability of a convex combination of the subsystems matrices.