Neumann-Rosochatius system for (m,n) string in $AdS_3 \times S^3$ with mixed flux

$SL(2,\mathbb{Z})$ 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 $(\cal{J})$.