This Note presents the resolution of a differential system on the plane that translates a geometrical problem about isotropic deformations of area and length. The system stems from a probability study on deformed random fields [1], which are the composition of a random field with invariance properties defined on the plane with a deterministic diffeomorphism. The explicit resolution of the differential system allows to prove that a weak notion of isotropy of the deformed field, linked to its excursion sets, in fact coincides with the strong notion of isotropy. The present Note first introduces the probability framework that gave rise to the geometrical issue and then proposes its resolution.