  In this paper we consider a question on existence of double series by
generalized Walsh system, which are universal in weighted $L_\mu^1[0,1]^2$
spaces. In particular, we construct a weighted function $\mu(x,y)$ and a double
series by generalized Walsh system of the form $$\sum_{n,k=1}^\infty
c_{n,k}\psi_n(x)\psi_k(y)\ \ \mbox{with} \ \ \sum_{n,k=1}^\infty \left |
c_{n,k} \right|^q <\infty\ \mbox{for all}\ q>2,$$ which is universal in
$L_\mu^1[0,1]^2$ concerning subseries with respect to convergence, in the sense
of both spherical and rectangular partial sums.
