  In the present paper the three state Potts model with competing binary
interactions (with couplings $J$ and $J_p$) on the second order Bethe lattice
is considered. The recurrent equations for the partition functions are derived.
When $J_p=0$, by means of a construction of a special class of limiting Gibbs
measures, it is shown how these equations are related with the surface energy
of the Hamiltonian. This relation reduces the problem of describing the limit
Gibbs measures to find of solutions of a nonlinear functional equation.
Moreover, the set of ground states of the one-level model is completely
described. Using this fact, one finds Gibbs measures (pure phases) associated
with the translation-invariant ground states. The critical temperature is
exactly found and the phase diagram is presented. The free energies
corresponding to translations-invariant Gibbs measures are found. Certain
physical quantities are calculated as well.
