We consider the following system of nonlinear evolution equations: \[ \begin{aligned} \dot b_n & = b_n\Biggl(c_1(b_{n+ 1}- b_{n- 1})- c_2\Biggl(b_{n+ 1} \Biggl(\sum^2_{k= 0} b_{n+ k}\Biggr)- b_{n- 1} \Biggl(\sum^2_{k= 0} b_{n+ k}\Biggr)\Biggr)\Biggr),\tag{1}\\ b_n & = b_n(t),\quad t\in [0, T),\quad n\in \mathbb{Z};\quad c_1, c_2\in \mathbb{C};\;\cdot = d/dt.\end{aligned} \] This system was studied in [\textit{O. I. Bogoyavlenskij}, Math. USSR, Izv. 32, No. 2, 245-268 (1989; Zbl 0672.35073)], where it was established that in the continuous limit it becomes the KdV equation for \(c_1\neq 12c_2\) and if \(c_1= 12c_2\) it becomes the second equation in the hierarchy of higher KdV equations. For \(c_1= 1\), \(c_2= 0\) system (1) is the well-known Volterra model. In the present note we give a procedure for constructing a solution of (1) from bounded initial data \((b_n(0))^\infty_{n= -\infty}\) as a power series in the time \(t\) using the method of the inverse spectral problem and its non-selfadjoint analog.