Motivated by recent studies related to integrability of string motion in various backgrounds via analytical and numerical procedures, we discuss these procedures for a well known integrable string background $(AdS_5times S^5)_{eta}$. We start by revisiting conclusions from earlier studies on string motion in $(mathbb{R}times S^3)_{eta}$ and $(AdS_3)_{eta}$ and then move on to more complex problems of $(mathbb{R}times S^5)_{eta}$ and $(AdS_5)_{eta}$. Discussing both analytically and numerically, we deduce that while $(AdS_5)_{eta}$ strings do not encounter any irregular trajectories, string motion in the deformed five-sphere can indeed, quite surprisingly, run into chaotic trajectories. We discuss the implications of these results both on the procedures used and the background itself.
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