Due to the inertia symmetry the issue of whether electrostatic solitons may possibly form in pair plasmas has been addressed in a number of papers. Recently we have shown that pair solitons with interlacing electron and positron holes in electron-positron plasmas may form by means of streaming instability based on one-dimensional particle-in-cell simulations Jao and Hau (2012 Phys. Rev. E 86 056401). In this paper we present the first simulation results of two-dimensional electrostatic solitons in pair plasmas imbedded in a stationary background magnetic field. It is shown that the features of electrostatic solitary structures may depend on the ratio of to where and denote the electron cyclotron and plasma frequency, respectively. In particular, for weakly magnetized or unmagnetized plasmas with both parallel and transverse electric field with the same order of magnitude may first develop and be dissipated in the nonlinear stage such that electrostatic solitons are unable to form by the streaming instability, while for pair solitons resembling those occurring in one-dimensional simulations may possibly form. Comparisons between linear fluid theory and particle-in-cell simulations are made.