On the sos model with one-level competing interactions on a binary tree

Remark 1. The system (3) coincides with the classical result for SOS model (see [2,5]) when , i.e.,

We have the following result:
Theorem 1. There is a bijection between the solutions of the system of nonlinear equations (3) and translation-invariant Gibbs measures.
References.
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2. Rozikov U. A. Gibbs measures on Cayley trees. // World scientific. – 2013.
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4. Ganikhodjaev N.N., Pah C.H. ,Wahiddin M.R. An Ising model with three competing interactions on a Cayley tree.// Journal of Math. Phys. –2004. Том 45.– С. 3645-3658.
5. Rozikov U.A., Suhov Yu.M. Gibbs measures for SOS model on a Cayley tree.// Infin. Dimens. Anal. Quantum Probab. Relat. Top. –2006 Том 9. № 3. – С. 471-488.