The collective elementary excitations of the two-dimensional (2D) magnetoexcitons in the state of their Bose-Einstein condensation (BEC) with nonzero wave vector k and inplane parallel oriented motional dipole moments are investigated in the Hartree-Fock-Bogoliubov approximation (HFBA). The breaking of the gauge symmetry is achieved using the Bogoliubov theory of quasiaverages and the Keldysh-Kozlov-Kopaev (KKK) method. The starting Hamiltonian and the Green's functions are determined using the integral two-particle operators instead of the single-particle Fermi operators. The infinite chains of equations of motion for the multioperator four- and six-particle Green-s functions are truncated following the Zubarev method and introducing a small parameter of the perturbation theory related with the lowest Landau levels (LLLs) filling factor and with the phase-space filling factor. The energy spectrum of the collective elementary excitations consists of the mixed exciton-plasmon energy braches, mixed exciton-plasmon quasienergy branches as well as the optical and acoustical plasmon energy branches. The exciton branches of the spectrum have gaps related with the negative values of the chemical potential and attractive interaction between the 2D megnetoexcitons with inplane, parallel oriented motional dipole moments. The slopes of the mixed exciton-plasmon branches are determined by the group velocities of the moving condensed excitons in the laboratory reference frame. The acoustical and optical plasmon energy branches are gapless. Their dependence on the small wave vectors accounted from the condensate wave vector k is linear and quadratic, respectively, with saturation in the range of high values of the wave vectors.