The singlet potential energy surfaces for all group 13 X2H-2 systems have been explored through ab initio SCF + CI calculations. Effective core potentials including relativistic effects for the heaviest atoms of the series were used. Geometries of the various isomers were determined at the Hartree-Fock level and confirmed to be minima by vibrational analysis. In all cases but boron, the global minimum is found to be the D2h di-H-bridged structure. For boron, the H-X=X-H linear form is found to be the global minimum, the 1DELTA(g) state being 14.7 kcal/mol above the 3SIGMA(g)- state. For all other atoms, this linear form is only a transition state, and a trans-bent isomer appears as a minimum on the surfaces for Al, Ga, and In. However, our analysis reveals two other low-lying minima, namely, the asymmetric X-XH-2 isomer and the C(s) mono-H-bridged structure. A simple rule for the occurrence of trans-bent isomers is derived from a MO model treating sigma-pi mixing. In HX=XH, a trans-bent distortion occurs when the singlet-triplet energy separation of the XH fragments is larger than half of the bond energy in the linear 1DELTA(g) state. Qualitative rules concerning the existence of the bridged structures are also established.