  $SL(2,\mathbb{Z})$ invariant action for probe $(m,n)$ string in $AdS_3\times
S^3\times T^4$ with mixed three-form fluxes can be described by an integrable
deformation of an one-dimensional Neumann-Rosochatius (NR) system. We present
the deformed features of the integrable model and study general class of
rotating and pulsating solutions by solving the integrable equations of motion.
For the rotating string, the explicit solutions can be expressed in terms of
elliptic functions. We make use of the integrals of motion to find out the
scaling relation among conserved charges for the particular case of constant
radii solutions. Then we study the closed $(m,n)$ string pulsating in
$R_t\times S^3$. We find the string profile and calculate the total energy of
such pulsating string in terms of oscillation number $(\cal{N})$ and angular
momentum $(\cal{J})$.
