The article is devoted to studying a linear difference equation of the form \[ (V\Omega)(z) \equiv \Omega(z - 1) + \Omega(z + 1) + G(z)\big(\Omega(z - i) + \Omega(z + i)\big) = g(z),\quad z \in D, \] under the following assumption: \(D\) is a square with vertices \(t_1 = - t_3 = -(1+i)/2\), \(t_2 = - t_4 = (1-i)/2\) and edges \(l_j\), \(j = 1,\dots, 4\), such that \(\{t_1,t_2\} \subset\bar l_1\); the coefficients \(G(z)\), \(g(z)\) are holomorphic on \(D\) and their boundary values \(G^+(t)\), \(g^+(t)\) belong to \(H(\Gamma)\) (i.e., satisfy the Hölder condition on \(\Gamma\)); \(G(z) \neq 0\) for all \(z\in\overline D\) and \(G(t_j) = 1\); solutions \(\Omega(z)\) are to be found in the class of functions holomorphic outside \(\overline D\) and vanishing at infinity; boundary values \(\Omega^-(t)\) belong to \(H(l_j)\), and only logarithmic singularities are allowed at the vertices. The case of \(G(z) \equiv 1\) was studied by the author in [Izv. Vyssh. Uchebn. Zaved., Mat. 1993, No. 7 (374), 7-16 (1993; Zbl 0839.45002)]. To solve the general case, the following integral representation of the equation is used: \[ \Phi(z) \equiv \frac{1}{2\pi i}\int_{\Gamma}\varphi(\tau)E(z,\tau) d\tau = g_1(z), \quad z \in D, \] where \(E(z,\tau) = G_1(z)[(\tau - z - 1)^{-1} + (\tau - z + 1)^{-1}] + G_2(z)[(\tau - z - i)^{-1} + (\tau - z + i)^{-1}]\). As a result, the author obtains the integral equation \[ (T\varphi)(t) \equiv \varphi(t) + \frac{1}{2\pi i\gamma_t}\int_{\Gamma} K(t,\tau)\varphi(\tau) d\tau = \frac{1}{\gamma_t}\big(g_{1}^+(t) - g_1^+(\alpha(t)) \big) \] with kernel \[ K(t,\tau) = E(t,\tau) - E\big(\alpha(t),\alpha(\tau)\big), \] where \(\alpha(t)\:\Gamma \to \Gamma\) denotes a shift operator that changes the orientation, and \(\gamma_t = \{G_1(t), \;t\in l_2\cup l_4;\;G_2(t), \;t\in l_1\cup l_3\}\) implies \(\gamma_t = \gamma_{\alpha(t)}\). The main result of the article reads as follows: Let the fundamental system of solutions to the adjoint equation \[ (T'\psi)(t) = 0 \] consists of \(m\) functions \(\gamma_t\), \(\psi_1(t)\), \(\psi_2(t),\dots,\psi_{m-1}(t)\) such that \(\gamma_t\), \(\psi_i(t) \in H(\bar l_j)\) and \(\gamma_t = \gamma(\alpha(t))\), \(\psi_i(t) = \psi_i(\alpha(t))\). Then, the original difference equation has exactly \(m\) solvability conditions.