  We probe warped BTZ $ \times S^3 $ geometry with various string solitons and
explore the classical integrability criteria of the associated phase space
configurations using Kovacic's algorithm. We consider consistent truncation of
the parent sigma model into one dimension and obtain the corresponding normal
variational equations (NVE). Two specific examples have been considered where
the sigma model is reduced over the subspace of the full target space geometry.
In both examples, NVEs are found to possess the Liouvillian form of solutions
which ensures the classical integrability of the associated phase space
dynamics. We address similar issues for the finite temperature counterpart of
the duality, where we analyze the classical phase space of the string soliton
probing warped BTZ black string geometry. Our analysis reveals clear
compatibility between normal variational equations and the rules set by the
Kovacic's criteria. This ensures the classical integrability of the parent
sigma model for the finite-temperature extension of the duality conjecture.
