Solitons in Kerr media with two-dimensional non-parity-time-symmetric complex potentials

•Two-dimensional (2D) non-parity-time (PT)-symmetric complex potentials have completely real spectra.•Continuous families of solitons can be stable in the 2D non-PT-symmetric complex potentials.•Eigenvalues of linear-stability spectra of solitons in 2D non-PT-symmetric complex potentials emerge as complex conjugate pairs.•Fundamental solitons bifurcate from the largest discrete eigenvalue of the linear spectrum.•Out-of-phase dipole solitons bifurcate from the second- or third-largest discrete eigenvalue of the linear spectrum. Our results indicate that continuous soliton families can exist and be stable in Kerr media with two-dimensional (2D) non-parity-time (PT)-symmetric complex potentials. There are several discrete eigenvalues in the linear spectra of these complex potentials. Fundamental solitons bifurcate from the largest discrete eigenvalue while the dipole solitons bifurcate from the second- or third- largest discrete eigenvalue. We further find that eigenvalues of the soliton linear-stability spectra emerge as complex conjugate pairs. The effect of different parameters of the complex potentials on soliton stability is discussed in detail. Moreover, we study the transverse energy flow vector of the solitons in these complex potentials.