  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 $(AdS_5\times
S^5)_{\eta}$. 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.
