Let $G$ be a locally compact abelian group and let $f: G \to \mathbb{R}^+$ be a continuous function satisfying the functional equation
$$f(x + \phi(y)) + f(y + \phi(x)) = 1 + f(x+y)$$
for all $x, y \in G$, where $\phi: G \to G$ is a continuous automorphism.
Suppose $\phi$ has finite order, i.e., there exists a positive integer $n$ such that $\phi^n(x) = x$ for all $x \in G$. What would be the structure of the solutions $f$ in this case?
