  Thin nematic elastomers, composite hydrogels and plant tissues are among many
systems that display uniform anisotropic deformation upon external actuation.
In these materials, the spatial orientation variation of a local director field
induces intricate global shape changes. Despite extensive recent efforts, to
date, there is no general solution to the inverse design problem: how to design
a director field that deforms exactly into a desired surface geometry upon
actuation, or whether such a field exists. In this work, we phrase this inverse
problem as a hyperbolic system of differential equations. We prove that the
inverse problem is locally integrable, provide an algorithm for its
integration, and derive bounds on global solutions. We classify the set of
director fields that deform into a given surface, thus paving the way to
finding optimized fields.
