There is Marangoni-thermocapillary convection of silicon melt in annular shallow pool submitted to bidirectional temperature gradients. When either of them exceeds the critical value, the flow becomes a three-dimensional oscillatory convection, which becomes complicated by thermal radiation on the free surface. In order to find an effective method to suppress the three-dimensional oscillatory convection, a series of three-dimensional numerical simulations, considering only pool rotation or only axial magnetic field or their coupling, are carried out. It is found that both pool rotation and axial magnetic field can suppress Marangoni-thermocapillary convection to a certain extent, and their combination can promote the suppression further. To avoid the negative effect of the magnetic field, the optimal parameters to maintain axisymmetric steady flow is obtained under the minimum magnetic field intensity.