One functional equation with displacement in the space of continuous functions

In the space of continuous functions defined on a simple continuous contour, we examine the functional equation
\begin{equation} a(t)\varphi(t)+b(t)\varphi[\alpha(t)]=g(t).\tag{1} \end{equation}
A criterion for Eq. (1) being Noetherian is established under the condition that there exist a finite number of fixed points on the first multiplicity in the homeomorphism $\alpha(t)$ of the contour onto itself.