The eigenvalues of the problems of \(\mathbb{R}\)-linear conjugation (Q1333667)

The author considers the problem of \(\mathbb{R}\)-linear conjugation \(\varphi^ +(t) = \varphi^ -(t) + \lambda \overline {\varphi^ - (t)}\), \(t \in \nu\), \(\varphi^ -(\infty) = 0\), where \(\gamma\) is a Lyapunov closed contour in the complex plane and \(\varphi (z)\) is sectionally holomorphic with jumping curve \(\gamma\). It is proved that the problem has nontrivial solutions (or has eigenvalues) iff \(\gamma\) is not a circle.