In this paper, we leverage the $O(2) \times \mathbb Z$-equivariant Leray-Schauder degree and a novel characterization of the Burnside Ring $A(O(2) \times \mathbb Z_2)$ presented by Ghanem in \cite{ Ghanem1} to obtain $(\rm i)$ an existence result for non-radial solutions to the problem $-\Delta u = f(z,u) + Au$, $u|_{ \partial D} = 0$ and $(\rm ii)$ local and global bifurcation results for multiple branches of non-radial solutions to the one-parameter family of equations $-\Delta u = f(z,u) + \textbf{ A}(\alpha)u$, $u|_{ \partial D} = 0$, where $D$ is the planar unit disc, $u(z) \in \mathbb R^N$, $A : \mathbb R^N \rightarrow \mathbb R^N$ is an $N \times N$ matrix, $\textbf{ A}: \mathbb R \rightarrow L(\mathbb R^N)$ is a continuous family of $N \times N$ matrices and $f: \overline D \times \mathbb R^N \rightarrow \mathbb R^N$ is a sublinear, $O(2) \times \mathbb Z_2$-equivariant function of order $o(|u|)$ as $u$ approaches the origin in $\mathbb R^N$.
Categories: math.AP