= Solution
In the order $(u,v,x,y)$ the <metric tensor> and <inverse metric> are
$$
(g_{ab})=
\begin{pmatrix}
F&1&0&0\\
1&0&0&0\\
0&0&1&0\\
0&0&0&1
\end{pmatrix},
\qquad
(g^{ab})=
\begin{pmatrix}
0&1&0&0\\
1&-F&0&0\\
0&0&1&0\\
0&0&0&1
\end{pmatrix}.
$$
Only $g_{uu}$ has nonzero <partial derivatives>. Substitution into the formula for the <Levi-Civita connection>,
$$
\Gamma^a{}_{bc}
=\frac12g^{ad}
(\partial_bg_{dc}+\partial_cg_{db}-\partial_dg_{bc}),
$$
therefore gives
$$
\boxed{
\Gamma^v{}_{uu}=\frac12F_{,u},\qquad
\Gamma^v{}_{ui}=\Gamma^v{}_{iu}=\frac12F_{,i},\qquad
\Gamma^i{}_{uu}=-\frac12F_{,i}.
}
$$
Every other <Christoffel symbol> vanishes. For example $g^{ud}$ selects $d=v$, and the <metric tensor> is independent of $v$ with constant $g_{av}$; hence $\Gamma^u{}_{bc}=0$.
Use the paper's <Riemann curvature tensor> convention
$$
R^d{}_{cab}
=\partial_a\Gamma^d{}_{bc}-\partial_b\Gamma^d{}_{ac}
+\Gamma^d{}_{ae}\Gamma^e{}_{bc}
-\Gamma^d{}_{be}\Gamma^e{}_{ac},
\qquad
R_{abcd}=g_{ae}R^e{}_{bcd}.
$$
For the independent components,
$$
R^v{}_{iuj}
=-\partial_j\Gamma^v{}_{ui}
=-\frac12F_{,ij}.
$$
The quadratic <Christoffel symbol> terms vanish here: any potentially contributing intermediate $u$ index gives a factor $\Gamma^u{}_{ab}=0$, and the remaining possible products have a lower $v$ index and vanish. Since the <symmetric matrix> $M$ gives $F_{,ij}=2M_{ij}$, lowering the first index gives
$$
\boxed{R_{uiuj}=R^v{}_{iuj}=-M_{ij}(u).}
$$
All remaining nonzero components follow from the <Riemann curvature tensor> symmetries
$$
R_{abcd}=-R_{bacd}=-R_{abdc}=R_{cdab}.
$$
There are no other independent components: terms involving a $v$ index vanish, while nonzero terms require one $u$ in each antisymmetric pair.
The <Ricci tensor> is consequently
$$
R_{uu}=\delta^{ij}R_{iuju}=-\operatorname{tr}M,\qquad
R_{ab}=0\quad\text{otherwise}.
$$
The <Ricci scalar> vanishes even when the <trace> is nonzero, since $g^{uu}=0$. With zero <cosmological constant>, the vacuum <Einstein field equations> reduce to $R_{ab}=0$. Therefore the necessary and sufficient condition is
$$
\boxed{\operatorname{tr}M(u)=0\quad\text{for every }u.}
$$
This is the <Ricci tensor of a Brinkmann plane wave>: a nonzero, trace-free profile can have nonzero <Riemann curvature tensor> despite vanishing <Ricci tensor>.
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