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Motivated by recent studies related to integrability of string motion invarious backgrounds via analytical and numerical procedures, we discuss theseprocedures for a well known integrable string background $(AdS_5\timesS^5)_{\eta}$. We start by revisiting conclusions from earlier studies on stringmotion in $(\mathbb{R}\times S^3)_{\eta}$ and $(AdS_3)_{\eta}$ and then move onto more complex problems of $(\mathbb{R}\times S^5)_{\eta}$ and$(AdS_5)_{\eta}$. Discussing both analytically and numerically, we deduce thatwhile $(AdS_5)_{\eta}$ strings do not encounter any irregular trajectories,string motion in the deformed five-sphere can indeed, quite surprisingly, runinto chaotic trajectories. We discuss the implications of these results both onthe procedures used and the background itself.

Probing analytical and numerical integrability: The curious case of $(AdS_5\times S^5)_{\eta}$