Helicoidal surfaces of prescribed mean curvature in $\mathbb{R}^3$

Given a function $\mathcal{H} \in C^1(\mathbb{S}^2)$, an $\mathcal{H}$-surface $\Sigma$ is a surface in the Euclidean space $\mathbb{R}^3$ whose mean curvature $H_\Sigma$ satisfies $H_\Sigma = \mathcal{H} \circ \eta$, where $\eta$ is the Gauss map of $\Sigma$. The purpose of this paper is to use a phase space analysis to give some classification results for helicoidal $\mathcal{H}$-surfaces, when $\mathcal{H}$ is rotationally symmetric, that is, $\mathcal{H} \circ \eta = \mathfrak{h} \circ \nu$, for some $\mathfrak{h} \in C^1([-1,1])$, where $\nu$ is the angle function of the surface. We prove a classification theorem for the case where $\mathfrak{h}(t)$ is even and increasing for $t \in [0,1]$. Finally, we provide examples of helicoidal $\mathcal{H}$-surfaces in cases where $\mathfrak{h}$ vanishes at some point.