We analyze the dynamics of a class of $\mathbb{Z}_6-$equivariant systems of the form $\dot{z}=pz^2\bar{z}+sz^3\bar{z}^2-\bar{z}^{5},$ where $z$ is complex, the time $t$ is real, while $p$ and $s$ are complex parameters. This study is the natural continuation of a previous work (M.J. \'Alvarez, A. Gasull, R. Prohens, Proc. Am. Math. Soc. \textbf{136}, (2008), 1035--1043) on the normal form of $\mathbb{Z}_4-$equivariant systems. Our study uses the reduction of the equation to an Abel one, and provide criteria for proving in some cases uniqueness and hyperbolicity of the limit cycle surrounding either 1, 7 or 13 critical points, the origin being always one of these points.