  In this paper we'll give a connection between four competing interactions
(external field, nearest neighbor, second neighbors and triples of neighbors)
of models with uncountable (i.e. $[0,1]$) set of spin values on the Cayley tree
of order two and Lyapunov integral equation. Also we'll study fixed points of
Lyapunov operator with degenerate kernel which each fixed point of the operator
is correspond to a {\it translation-invariant} Gibbs measure.
