Homological equations and topological properties of $S^1$-extensions over an ergodic rotation of the circle

A description is given of the set of $\beta\in[0;1]$, such that the homological equation
$$ f(x+\beta)-f(x)=g(x+\alpha)-g(x) $$
has a continuous solution, where $f(x)$ is a continuous periodic function, $f(x+1)=f(x)$. The result obtained is applied in studying the property of relative separability of $S^1$-extensions over an ergodic rotation of the circle.