  Motivated by the design of flexible nets, we classify all nets of arbitrary
size m x n that admit a continuous family of area-preserving Combescure
transformations. There are just two different classes. The nets in the first
class are special cases of cone nets that have been recently studied by Kilian,
Mueller, and Tervooren. The second class consists of Koenigs nets having a
Christoffel dual with the same areas of corresponding faces. We apply isotropic
metric duality to get a new class of flexible nets in isotropic geometry. We
also study the smooth analogs of the introduced classes.
