  A parametric family of two-dimensional random walks $\mathbf{S}_t(a)$
$=\big(S_t^{(1)}(a),$ $S_t^{(2)}(a)\big)$ in the main quarter plane is studied.
The components $S_t^{(1)}(a)$ and $S_t^{(2)}(a)$ are assumed to be correlated
in the way that is defined exactly in the paper.
  We derive the conditions on $a$, under which a random walk $\mathbf{S}_t(a)$
is recurrent.
