  We construct local solutions to 11-dimensional supergravity (or M-theory),
which are invariant under the superalgebra $D(2, 1; c'; 0)\oplus D(2, 1; c';
0)$ for all values of the parameter $c'$. The BPS constraints are reduced to a
single linear PDE on a complex function $G$. The PDE was solved in 0806.0605
modulo application of boundary and regularity conditions. The physical fields
of the solutions are determined by $c'$, a harmonic function $h$, and the
complex function $G$. $ h$ and $G$ are both functions on a 2-dimensional
compact Riemannian manifold. The harmonic function $ h$ is freely chosen. We
obtain the expressions for the metric and the field strength in terms of $G$,
$h$, and $c'$ and show that these are indeed valid solutions of the Einstein,
Maxwell, and Bianchi equations. Finally we give a construction of one parameter
deformations of $AdS_7 \times S^4$ and $AdS_4 \times S^7$ as a function of
$c'$.
