  We consider a (2+1)-dimensional Toda-like chain which can be viewed as a
two-dimensional generalization of the Wu-Geng model and which is closely
related to the two-dimensional Volterra, two-dimensional Toda and relativistic
Toda lattices. In the framework of the Hirota direct approach, we present
equations describing this model as a system of bilinear equations that belongs
to the Ablowitz-Ladik hierarchy. Using the Jacobi-like determinantal identities
and the Fay identity for the theta-functions, we derive its Toeplitz,
dark-soliton and quasiperiodic solutions as well as the similar set of
solutions for the two-dimensional Volterra chain.
