= Solution
For the given metric, direct substitution of the stated connection coefficients into the curvature formula leaves
$$
R_{uu}=-\frac12(H_{,xx}+H_{,yy}).
$$
The inverse metric has $g^{uu}=0$, and every other Ricci component vanishes, so
$$
R=g^{\alpha\beta}R_{\alpha\beta}=0.
$$
Thus the full, nonlinear vacuum Einstein equation is precisely the transverse Laplace equation from a <plane-fronted gravitational wave>:
$$
H_{,xx}+H_{,yy}=0.
$$
The dependence on $u$ is unrestricted.
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