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 Z/dZ, for any d≥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.
