The bound states of two interacting two-dimensional magnetoexcitons with electrons and holes on the lowest Landau levels (LLLs) moving in-plane of the layer with equal but opposite oriented wave vectors and forming a molecular-type structures with the resultant wave vector ^→ k = 0 were investigated. Four possible spin structures of two electrons and of two holes forming the bound states were considered. Two of them lead to the formation of the para and ortho magnetoexcitons in the presence of the electron-hole (e-h) Coulomb exchange interaction. In this case we have studied the interaction of two para magnetoexcitons and of two ortho magnetoexcitons with the resultant spin equal to zero. Another two variant, are actual when the Coulomb exchange e-h interaction is negligible small and the spin of two electrons separately and the effective spin of two holes are interconnected and forms the singlet or the triplet states with zero spin projections on the magnetic field direction. The spin states of the four particles were constructed combining the singlet two electron state with the singlet two hole state as well as the triplet two electron state with the triplet two hole state. Only the bound states of two electrons and of two holes with singlet-singlet and with triplet-triplet spin structures were studied. It was shown that the spin structure of the type singlet-triplet and triplet-singlet do not exist due to the hidden symmetry of the magnetoexcitons. The orbital structure of the 2D magnetoexciton with wave vector ^→ k ≠ 0 is similar with an in-plane electric dipole with the dipole moment perpendicularly oriented to the wave vector. The bimagnetoexciton with resultant wave vector ^→ k = 0 is composed from two antiparallel oriented electric dipoles moving with antiparallel wave vectors ^→ k ≠ 0. Their relative motion in the frame of the bound states is characterized by the variational wave functions φ _n ( ^→ k) depending on the modulus ^→ k. It was shown that the stable bound state in the lowest Landau levels approximation do not exist in four investigated spin combinations. Instead of them a deep metastable bound state with an activation barrier comparable with the ionization potential of the magnetoexciton with ^→ k = 0 was revealed in the triplet-triplet spin configuration. Its orbital structure in the momentum space representation is characterized by the maximal exciton density on the in-plane ring and with zero density in the center.