Integrable deformations of the flat space sigma model

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.