  A Mathieu-Zhao subspace is a generalization of an ideal of an associative
algebra $\mathcal A$ over a unital ring $R$ first formalized in 2010. A vertex
algebra is an algebraic structure first developed in conjunction with string
theory in the 1960s and later axiomatized by mathematicians in the 1980s. We
formally introduce the definition of a Mathieu-Zhao subspace $M$ of a vertex
algebra $V$. Building on natural connections to associative algebras, we
classify an infinite set of non-trivial, non-ideal Mathieu-Zhao subspaces for
simple and general vertex algebras by group action eigenspace decomposition.
Finally, we state the locally nilpotent $\varepsilon$-derivation (LNED)
conjecture for vertex algebras.
