Neumann-Rosochatius system for strings in ABJ Model

Neumann-Rosochatius system is a well known one dimensional integrable system. We study the rotating and pulsating string in $AdS_4 times mathbb{CP}^3$ with a $B_{rm{NS}}$ holonomy turned on over $mathbb{CP}^1 subset mathbb{CP}^3$, or the so called Aharony-Bergman-Jafferis (ABJ) background. We observe that the string equations of motion in both cases are integrable and the Lagrangians reduce to a form similar to that of deformed Neuman-Rosochatius system. We find out the scaling relations among various conserved charges and comment on the finite size effect for the dyonic giant magnons on $R_{t}times mathbb{CP}^{3}$ with two angular momenta. For the pulsating string we derive the energy as function of oscillation number and angular momenta along $mathbb{CP}^{3}$.