  In this paper, we study some aspects of the irreducibility of
$\widetilde{M_g^{Pl}(G)}$ and its interrelation with the existence of "normal
forms", i.e. non-singular plane equations (depending on a set of parameters)
such that a specialization of the parameters gives a certain non-singular plane
model associated to the elements of $\widetilde{M_g^{Pl}(G)}$. In particular,
we introduce the concept of being equation strongly irreducible
(ES-Irreducible) for which the locus $\widetilde{M_g^{Pl}(G)}$ is represented
by a single "normal form". Henn, and Komiya-Kuribayashi, observed that
$\widetilde{M_3^{Pl}(G)}$ is ES-Irreducible. In this paper we prove that this
phenomena does not occur for any odd $d>4$. More precisely, let
$\mathbb{Z}/m\mathbb{Z}$ be the cyclic group of order $m$, we prove that
$\widetilde{M_g^{Pl}(\mathbb{Z}/(d-1)\mathbb{Z})}$ is not ES-Irreducible for
any odd integer $d\geq5$, and the number of its irreducible components is at
least two. Furthermore, we conclude the previous result when $d=6$ for the
locus $\widetilde{M_{10}^{Pl}(\mathbb{Z}/3\mathbb{Z})}$.
  Lastly, we prove the analogy of these statements when $K$ is any
algebraically closed field of positive characteristic $p$ such that
$p>(d-1)(d-2)+1$.
