We establish 1d Neumann-Rosochatius (NR) integrable model for rotating and pulsating string ansatz in some less conventional 2d nonlinear string sigma models, specifically, the fundamental string probing the gravity background of planar ABJ theory and the manifest SL(2, Z) covariant (p,q)-type bound states of fundamental string and D1 branes in $AdS_3\times S^3\times T^4$ background with mixed NSNS-RR flux. The main idea of our work is to confront the spectral problem of quite complicated 2d string sigma-models in the light of an exactly solvable 1d model. With explicit formulations of Lagrangian, Hamiltonian and conserved integrals of motion, we verified that both of the above systems admit a systematic reduction into the NR integrable model in the presence of finite flux and other intricate background features. The energy spectra for our generic solutions are evaluated by solving the integrable equations of motion of the NR model. These are subsequently found to be consistent with some known local gauge-invariant operators as well as the relevant integrable spin chain descriptions, at least up to some special limits.