The electronic structures of non-Kekule-type isoelectronic molecules 1,3,5-trimethylenebenzene (TMB) and 1,3,5-triaminobenzene trication (TAB(3+)) having a threefold axis are discussed with the ab initio molecular orbital (MO) method. The quartet state with a D-3h geometry is predicted to be the ground state of TMB and TAB(3+). According to the Jahn-Teller theorem, the doublet (2)E" states of TMB and TAB(3+) are subject to the first order distortions which remove the degeneracy. The quartet-doublet energy splittings are;calculated for these nonKekule-type molecules at the second order Moller-Plesset (MP2) perturbation level of theory. In TMB the (4)A"(2) State with a planar D-3h geometry is predicted to lie well below the (2)A(2) State with a planar C-2 upsilon geometry in which One of the methylene groups has a longer C-C bridge bond connecting with the benzene ring. Moreover, in TAB(3+) the (4)A"(2) state with a planar D-3h geometry also lies well below the (2)A(2) state. The (2)A(2)-B-2(1) energy splittings on the pseudorotation potential surface overestimated at the unrestricted Hartree-Fock (UHF) level are corrected with the MP2 method.