  In this note the logarithmic energy problem with external potential
$|z|^{2n}+tz^d+\bar{t}\bar{z}^d$ is considered in the complex plane, where $n$
and $d$ are positive integers satisfying $d\leq 2n$. Exploiting the discrete
rotational invariance of the potential, a simple symmetry reduction procedure
is used to calculate the equilibrium measure for all admissible values of $n,d$
and $t$.
  It is shown that, for fixed $n$ and $d$, there is a critical value
$|t|=t_{cr}$ such that the support of the equilibrium measure is simply
connected for $|t|<t_{cr}$ and has $d$ connected components for $|t|>t_{cr}$.
