Connecting continua and curves of equilibria of the Cahn-Hilliard equation on the square

We state an alternative for paths of equilibria of the Cahn-Hilliard equation on the square, bifurcating from the trivial solution at eigenfunctions of the form $w_{ij}=\cos(\pi ix)\cos(\pi j y)$, for $i,j \in \N$. We show that the paths either only connect the bifurcation point $m_{ij}$ with $-m_{ij}$ and are separated from all other paths with even more symmetry, or they contain a loop of nontrivial solutions connecting the bifurcation point $m_{ij}$ with itself. In any case the continua emerging at $m_{ij}$ and $-m_{ij}$ are equal. For fixed mass $m_0=0$ we furthermore prove that the continua bifurcating from the trivial solution at eigenfunctions of the form $w_{i0}+w_{0i}$ or $w_{ij}$, for $i,j \in \N$ are smooth curves parameterized over the interaction length related parameter $\lambda$.