  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.
