= Solution
Stationarity changes $\Box$ to the spatial <Laplace operator>. The Green function of the <Poisson equation> therefore gives
$$
\boxed{\bar h_{00}(\mathbf r)=4\int_{\mathbb R^3}
\frac{\rho(\mathbf r')}{|\mathbf r-\mathbf r'|}\,d^3r'},
$$
$$
\boxed{\bar h_{0i}(\mathbf r)=4\Omega\int_{\mathbb R^3}
\frac{\rho(\mathbf r')}{|\mathbf r-\mathbf r'|}
(y',-x',0)_i\,d^3r'},
\qquad
\boxed{\bar h_{ij}=0}.
$$
Here the bar denotes the <trace-reversed metric perturbation>; this is the variable denoted by $h$ in the displayed field equation of the question.
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