Symmetries for Julia sets for perturbations of polynomial maps

This is a naive question. Consider the Julia sets of the map $ $ z \mapsto z^n + \lambda / z^k $ $ with $ z,\lambda \in \mathbb{C}$ , and the exponents $ n,k \in \mathbb{N}$ . For example, for $ n=k=3$ , here are three Julia sets depictions for three different (relatively small) $ \lambda$ ‘s:

          z^3 & lambdas
          L-to-R:    $ \lambda=0.2-0.1 i$ ;   $ \lambda=2-i$ ;    $ \lambda=0.07-0.5i$ .

There is obvious hexagonal symmetry independent of $ \lambda$ . My question is:

Q. Given $ n$ and $ k$ , are the symmetries of the Julia sets of this map, for relatively small $ \lambda$ , known? Should they always have $ (n+k)$ -gon-like symmetries?

I can imagine that many details of these maps are unknown, but perhaps at this high-level viewpoint, the gross structure of the Julia sets is known? So for $ n=4$ and $ k=5$ , we should see nonagonal symmetries?