  We explore analytic integrability criteria for $ D1 $ branes probing 4D
relativistic background with a null isometry direction. We use both the
Kovacic's algorithm of classical (non)integrability as well as the standard
formulation of Lax connections to show the analytic integrability of the
associated dynamical configuration. We further use the notion of double null
reduction and obtain the world-volume action corresponding to a torsional
Newton-Cartan (TNC) $ D0 $ brane probing a 3D torsional Newton-Cartan geometry.
Moreover, following Kovacic's method, we show the classical integrability of
the TNC $ D0 $ brane configuration thus obtained. Finally, considering a
trivial field redefinition for the $ D1 $ brane world-volume fields, we show
the equivalence between two configurations in the presence of vanishing NS
fluxes.
