  We consider the most general solutions of eleven-dimensional supergravity
preserving $N=2$ supersymmetry whose metrics are warped products of
three-dimensional anti-de Sitter space with an eight-dimensional manifold,
focusing on those realising (2,0) superconformal symmetry. We give a set of
necessary and sufficient conditions for a solution to be supersymmetric, which
can be phrased, in the general case, in terms of a local SU(2) structure and
its intrinsic torsion. We show that these supergravity backgrounds always admit
a nowhere-vanishing Killing vector field that preserves the solution and
encodes the U(1) R-symmetry of the dual field theory. We illustrate our results
with examples which have appeared in the literature, including those with
SU(4), G$_2$ and SU(3) structures, and discuss new classes of Minkowski
solutions.
