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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 asingle linear PDE on a complex function $G$. The PDE was solved in 0806.0605modulo application of boundary and regularity conditions. The physical fieldsof the solutions are determined by $c'$, a harmonic function $h$, and thecomplex function $G$. $ h$ and $G$ are both functions on a 2-dimensionalcompact Riemannian manifold. The harmonic function $ h$ is freely chosen. Weobtain 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 parameterdeformations of $AdS_7 \times S^4$ and $AdS_4 \times S^7$ as a function of$c'$.

Exact Half-BPS Flux Solutions in M-theory with $D(2, 1; c'; 0)^2$ Symmetry: Local Solutions