Analytic integrability for strings on $ \eta $ and $ \lambda $ deformed backgrounds

In this paper, based on simple analytic techniques, we explore the integrability conditions for classical stringy configurations defined over $\eta$ as well as $\lambda$- deformed backgrounds. We perform our analysis considering classical string motions within various subsectors of the full target space geometry. It turns out that classical string configurations defined over $\eta$- deformed backgrounds are non-integrable whereas on the other hand, the corresponding configurations are integrable over the $\lambda$- deformed background. Our analysis therefore imposes a strong constraint on the operator spectrum associated with the corresponding dual gauge theories at strong coupling.