  We study the homogeneous artinian ideals of the polynomial ring $K[x,y,z]$,
generated by the homogenous polynomials of degree $d$ which are invariant under
an action of the cyclic group $\mathbb Z/d\mathbb Z$, for any $d\geq 3$. We
prove that they are all monomial Togliatti systems, and that they are minimal
if the action is defined by a diagonal matrix having on the diagonal $(1, e,
e^a)$, where $e$ is a primitive $d$-th root of the unity. We get a complete
description when $d$ is prime or a power of a prime. We also establish the
relation of these systems with linear Ceva configurations.
