In this study, the MHD effect on boundary layer rotating flow of a nanofluid is investigated for the multiple branches case. The main focus of current research is to examine flow characteristics on a nonlinear permeable shrinking sheet. Moreover, the governing partial differential equations (PDEs) of the problem considered are reduced into coupled nonlinear ordinary differential equations (ODEs) with the appropriate similarity transformation.  Numerical results based on the plotted graphs are gotten by solving ODEs with help of the three-stage Labatto IIIA method in bvp4c solver in MATLAB. To confirm numerical outcomes, current results are compared with previously available outcomes and found in good agreement. Skin friction coefficients, Nusselt and Sherwood numbers, velocity profiles, temperature profiles, and concentration profiles are examined. The results show that dual (no) branches exist in certain ranges of the suction parameter i.e., SSc (SSc). Further, profiles of velocity decrease for rising values of Hartmann number in the upper branch, while reverse trend has been noticed in a lower branch. Profiles of temperature and concentration enhance for the increasing values of thermophoresis in both branches. stability analysis of the branches is also done and noticed that upper branch is a stable branch from both branches. Finally, it is noted that the stable branch has symmetrical behavior with regard to the parameter of rotation.