  In this paper we study the existence of periodic travelling waves for the 2D
$abcd$ Boussinesq type system related with the three-dimensional water-wave
dynamics in the weakly nonlinear long-wave regime. We show that small solutions
that are periodic in the direction of translation (or orthogonal to it) form an
infinite-dimensional family, by characterizing these solutions through spatial
dynamics when we reduce to a center manifold of infinite dimension and
codimension the linearly ill-posed mixed-type initial-value problem. As happens
for the Benney-Luke model and the KP II model for wave speed large enough and
large surface tension, we show that a unique global solution exists for
arbitrary small initial data for the two-component bottom velocity, specified
along a single line in the direction of translation (or orthogonal to it). As a
consequence of this fact, we show that the spatial evolution of bottom velocity
is governed by a dispersive, nonlocal, nonlinear wave equation
