Singular field redefinition between Witten's string field theory and Witten's theory deformed by Ellwood invariant

We construct a field redefinition between Witten’s string field theory and its deformation by the Ellwood invariant. This field redefinition is singular and does not imply physical equivalence between them. However, it allows us to formally transfer classical solutions of Witten’s theory to solutions of the deformed theory. Although the resulting solutions are also generically singular and require careful examination of their physical interpretation, we show that the tachyon vacuum solution can be consistently transferred from Witten’s theory to the deformed theory.