  The purpose of this work is to close the local deformation problem of rank
two Euclidean submanifolds in codimension two by describing their moduli space
of deformations. In the process, we provide an explicit simple representation
of these submanifolds, a result of independent interest by its applications. We
also determine which deformations are genuine and honest, allowing us to find
the first known examples of honestly locally deformable rank two submanifolds
in codimension two. In addition, we study which of these submanifolds admit
isometric immersions as Euclidean hypersurfaces, a property that gives rise to
several applications to the Sbrana-Cartan theory of deformable Euclidean
hypersurfaces.
