In this paper, we provide both a preservation and breaking of symmetry theorem for $2pi$-periodic problems of the form begin{align*} begin{cases} -u(t) + g(u(t)) = f(t)cr u(0) - u(2pi) = u(0) - u(2pi) = 0 end{cases} end{align*} where $g: mathbb{R} to
mathbb{R}$ is a given $C^1$ function and $f: [0,2pi] to mathbb{R}$ is continuous. We provide a preservation of symmetry result that is analogous to one given by Willem (Willem, 1989) and a generalization of the theorem given by Costa-Fang (Costa and Fang, 2019). Both of these theorems use group actions that are not normally considered in the literature.
