= Solution
On a bosonic background, a <Killing spinor> is a nonzero spinor parameter that makes all fermionic <supersymmetry> variations vanish. It identifies an unbroken supersymmetry of the background; the number of independent solutions counts preserved supercharges. In simple ungauged <minimal four-dimensional supergravity>, with the gravitino set to zero, its equation is $\nabla_\mu\epsilon=0$ and the bosonic field equations are $R_{\mu\nu}=0$. In theories with additional fields or a cosmological deformation, the Killing-spinor equation instead uses the corresponding modified connection.
Use commuting spinors to form ordinary geometric bilinears, and orient the <Dirac current> to the future with $\bar\epsilon=-\epsilon^\dagger\gamma^{\hat0}$ in mostly-plus signature. Since the spinor and gamma matrices are parallel,
$$
\nabla_\mu K_\nu=(\nabla_\mu\bar\epsilon)\gamma_\nu\epsilon+\bar\epsilon\gamma_\nu\nabla_\mu\epsilon=0.
$$
Thus $K$ is actually parallel, and in particular is a <Killing vector field>. In a local orthonormal frame,
$$
K^{\hat0}=\epsilon^\dagger\epsilon>0,\qquad
K^{\hat i}=\epsilon^\dagger\alpha^i\epsilon,\qquad\alpha^i=-\gamma^{\hat0}\gamma^{\hat i}.
$$
For every spatial unit vector $u$, $u_i\alpha^i$ is Hermitian and squares to one. Its eigenvalues are $\pm1$, so $|u_iK^i|\leq K^0$. Taking $u$ along the spatial current gives
$$
\boxed{K_\mu K^\mu=-(K^0)^2+|\mathbf K|^2\leq0.}
$$
A nonzero parallel spinor cannot vanish at an isolated point, so this causal <Dirac current> is nonzero everywhere on a connected solution. Its squared norm is constant: it cannot alternate between timelike and null regions.
If $K$ is timelike, normalize its constant norm. Its parallel dual one-form is closed, and its orthogonal distribution is integrable. Adapted local coordinates put the metric in the product form
$$
ds^2=-dt^2+h_{ij}(x)dx^idx^j.
$$
The time independence follows because $K$ is Killing, and the unit lapse and zero cross terms follow from parallelism. The vacuum equations give $R_{ij}(h)=0$. In three dimensions the Riemann tensor is completely determined by the <Ricci tensor> and its scalar contraction, so $h$ is flat. Therefore
$$
\boxed{\text{Every timelike-current solution is locally Minkowski spacetime.}}
$$
Constant spinors in its flat frame provide all local Killing spinors. Global identifications are possible only when they preserve a compatible <spin structure> and the required spinors.
If $K$ is null, its parallelism first gives a Brinkmann coordinate system with $K=\partial_v$. The parallel-spinor condition further restricts the local spin holonomy to the stabilizer of that spinor. In four dimensions, for a nonzero Weyl component represented by $(1,0)^T$, its stabilizer in $SL(2,\mathbb C)$ is the upper-triangular group $\begin{pmatrix}1&b\\0&1\end{pmatrix}$, $b\in\mathbb C$: the two real null rotations, with no transverse rotation. Consequently the transverse frame is flat and the allowed curvature is of the wave type $R_{uiuj}$. Using the coordinate freedom to remove transverse cross terms gives the local <plane-fronted gravitational wave> form
$$
\boxed{ds^2=-2du\,dv+dx^2+dy^2+H(u,x,y)du^2,\qquad H_{xx}+H_{yy}=0.}
$$
Here $R_{uiuj}=-H_{ij}/2$ and the only potentially nonzero Ricci component is $R_{uu}=-(H_{xx}+H_{yy})/2$, so the last equation is exactly the vacuum equation. Conversely, in the coframe $e^+=du$, $e^-=dv-Hdu/2$, $e^i=dx^i$, the nonzero spin-connection matrices are proportional to $H_i\gamma^{+i}du$. Constant spinors with $\gamma^+\epsilon=0$ are therefore parallel. Every harmonic transverse profile, with arbitrary $u$ dependence, gives such a supersymmetric vacuum wave; the flat profile is included. This verifies both the form and sufficiency in the <parallel-spinor classification of vacuum four-geometries>.
There is a reality-condition subtlety in interpreting the branches. A single commuting four-dimensional Majorana spinor has a <null current of a commuting Majorana spinor>: its two-component representation gives $|\xi^\dagger\boldsymbol\sigma\xi|^2=(\xi^\dagger\xi)^2$. A timelike current can instead come from a Dirac Killing spinor or a sum of independent real-spinor currents. The conditional timelike classification above remains valid, but does not assert a generic timelike current for a single real Majorana parameter.
For the final question use constant sectional curvature $-1/a^2$, so $R_{\mu\nu}= -3g_{\mu\nu}/a^2$. The <spinor curvature identity> gives
$$
[\nabla_\mu,\nabla_\nu]\epsilon=-\frac1{2a^2}\gamma_{\mu\nu}\epsilon.
$$
Define the <Anti-de Sitter Killing-spinor connection> $\mathcal D_\mu=\nabla_\mu+\gamma_\mu/(2a)$. Since $\nabla_\mu\gamma_\nu=0$ and the connection is torsion-free,
$$
[\mathcal D_\mu,\mathcal D_\nu]
=-\frac1{2a^2}\gamma_{\mu\nu}+\frac1{4a^2}[\gamma_\mu,\gamma_\nu]=0.
$$
Thus the modified connection is flat. On a simply connected region, transport any initial spinor with this connection; flatness makes the result path independent and supplies actual nonzero solutions, not merely a necessary integrability condition. Hence
$$
\boxed{\nabla_\mu\epsilon+\frac1{2a}\gamma_\mu\epsilon=0}
$$
has a full local space of solutions on <Anti-de Sitter spacetime>, and globally on its universal cover. Global quotients require compatible spinor monodromy. These are Killing spinors of the cosmological deformation, not parallel-spinor vacua of the undeformed zero-cosmological-constant theory.
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