  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
[J.Fournier '17], 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.
