  We explore a deformation of the flat space symmetric space sigma model
action. The deformed action is designed to allow a Lax connection for the
equations of motion, similar to the undeformed model. For this to work, we
identify a set of constraints that the deformation operator, which is
incorporated into the action, must fulfil. After defining the deformation, we
explore simple solutions to these constraints and describe the resulting
deformed backgrounds. Specifically, we find flat space in Cartesian coordinates
with arbitrary constant $H$-flux or linear $H$-flux in a light cone coordinate.
Additionally, we find the Nappi-Witten background along with various
Nappi-Witten-like backgrounds with near arbitrary constant $H$-flux. Finally,
we discuss the symmetries of the deformed models, finding that the deformed
symmetries will always include a set of symmetries that in the undeformed limit
becomes the total set of translations.
