  In this paper, we prove the existence of non-radial solutions to the problem
$-\triangle u=f(z,u)$, $u|_{\partial D}=0$ on the unit disc $D:=\{z\in \mathbb
C : |z|<1\}$ with $u(z)\in \mathbb R^k$, where $f$ is a sub-linear continuous
function, differentiable with respect to $u$ at zero and satisfying
$f(e^{i\theta}z,u) = f(z,u)$ for all $\theta\in \mathbb R$, $f(z,-u)=- f(z,u)$.
Under the assumption that $f$ respects additional (spacial) symmetries on
$\mathbb R^k$, we investigate symmetric properties of the corresponding
non-radial solutions. The abstract result is supported by a numerical example
with extra $S_4$-symmetries.
