We consider a Schrödinger operator \(H_\mu (K)\) corresponding to the Hamiltonian of the system of three identical particles on three dimensional lattice with attracting contact potentials, where \(K\in {\mathbb {T}}^{3}\) is a quasi-momentum, \({\mathbb {T}}^{3}\) is a three dimensional torus, \(\mu \) is an interaction energy of two particles. We prove the existence of a unique \(\mu =\mu _0\) such that the discrete spectrum of \(H_{\mu _0}({\textbf {0}})\) is empty and the bottom of essential spectrum is an eigenvalue of \(H_{\mu _0}({\textbf {0}})\) . For this case we show that for all nontrivial quasi-momentum \(K\in {\mathbb {T}}^{3} \) discrete spectrum of \(H_{\mu _0}(K)\) is non-empty.
