## Abstract.

We prove a rigidity result for pairs of mappings of integrable dilatation whose gradients pointwise deform the unit ball to similar ellipses. Our result implies as corollaries a version of the generalized Stoilow decomposition provided by Theorem 5.5.1 of [Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane, Princeton University Press, Princeton, 2009] and the two-dimensional rigidity result of [Calc. Var. Partial Differential Equations (2012), DOI 10.1007/s00526-012-0566-4] for mappings whose symmetric part of gradient agrees.

Specifically let $u,v\in {W}^{1,2}(\Omega ,{\mathbb{R}}^{2})$ where $det\left(Du\right)>0$, $det\left(Dv\right)>0$ a.e. and *u* is a mapping of integrable dilatation. Suppose for
a.e. $z\in \Omega $ we have
$Du{\left(z\right)}^{T}Du\left(z\right)=\lambda Dv{\left(z\right)}^{T}Dv\left(z\right)\phantom{\rule{1.em}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{some}\phantom{\rule{4.pt}{0ex}}\lambda >0\text{.}$
Then there exists a meromorphic function ψ and a homeomorphism $w\in {W}^{1,1}(\Omega :{\mathbb{R}}^{2})$ such that
$Du\left(z\right)=\mathcal{P}\left(\psi \left(w\right(z\left)\right)\right)Dv\left(z\right)$
where
$\mathcal{P}(a+ib)=\left(\genfrac{}{}{0pt}{}{a\phantom{\rule{1.em}{0ex}}-b}{b\phantom{\rule{1.em}{0ex}}a}\right).$

We show by example that this result is sharp in the sense that there can be no continuous relation between the gradients of *D**u* and *D**v* on a dense open connected subset of Ω unless one of the mappings is of integrable dilatation.

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