  A deterministic application $\theta\,:\,\mathbb{R}^2\rightarrow\mathbb{R}^2$
deforms bijectively and regularly the plane and allows to build a deformed
random field $X\circ\theta\,:\,\mathbb{R}^2\rightarrow\mathbb{R}$ from a
regular, stationary and isotropic random field
$X\,:\,\mathbb{R}^2\rightarrow\mathbb{R}$. The deformed field $X\circ\theta$ is
in general not isotropic, however we give an explicit characterization of the
deformations $\theta$ that preserve the isotropy. Further assuming that $X$ is
Gaussian, we introduce a weak form of isotropy of the field $X\circ\theta$,
defined by an invariance property of the mean Euler characteristic of some of
its excursion sets. Deformed fields satisfying this property are proved to be
strictly isotropic. Besides, assuming that the mean Euler characteristic of
excursions sets of $X\circ\theta$ over some basic domains is known, we are able
to identify $\theta$.
