  We analyze a three-state Potts model built over a lattice ring, with coupling
$J_0$, and the fully connected graph, with coupling $J$. This model is
effectively mean-field and can be exactly solved by using transfer-matrix
method and Cardano formula. When $J$ and $J_0$ are both ferromagnetic, the
model has a first-order phase transition which turns out to be a smooth
modification of the known phase transition of the traditional mean-field Potts
model ($J_0=0$), despite, as we prove, the connected correlation functions are
now non zero, even in the paramagnetic phase. Furthermore, besides the
first-order transition, there exists also a hidden continuous transition at a
temperature below which the symmetric metastable state ceases to exist. When
$J$ is ferromagnetic and $J_0$ antiferromagnetic, a similar antiferromagnetic
counterpart phase transition scenario applies. Quite interestingly, differently
from the Ising-like two-state case, for large values of the antiferromagnetic
coupling $J_0$, the critical temperature of the system tends to a finite value.
