  In this paper we complete the analysis of a statistical mechanics model on
Cayley trees of any degree, started in
[EsHaRo12,EsRo10,BoEsRo13,JaKuBo14,Bo17]. The potential is of nearest-neighbor
type and the local state space is compact but uncountable. Based on the system
parameters we prove existence of a critical value $\theta_{\rm c}$ such that
for $\theta\le \theta_{\rm c}$ there is a unique translation-invariant
splitting Gibbs measure. For $\theta_{\rm c}<\theta$ there is a phase
transition with exactly three translation-invariant splitting Gibbs measures.
The proof rests on an analysis of fixed points of an associated non-linear
Hammerstein integral operator for the boundary laws.
