  This paper presents an $\mathcal{E}$-derivation analogue of a result on
derivations due to van den Essen, Wright and Zhao. We prove that the image of a
locally finite $K$-$\mathcal{E}$-derivation of polynomial algebras in two
variables over a field $K$ of characteristic zero is a Mathieu-Zhao subspace.
This result together with that of van den Essen, Wright and Zhao confirms the
LFED conjecture in the case of polynomial algebras in two variables.
