Starting from the axial heavy meson exchange currents, constructed earlier inconjunction with the Bethe--Salpeter equation, we first present the axial$\rho$--, $\omega$-- and $a_1$ meson exchange Feynman amplitudes that satisfythe partial conservation of the axial current. Employing these amplitudes, wederive the corresponding weak axial heavy meson exchange currents in theleading order in the 1/M expansion ($M$ is the nucleon mass), suitable for thenuclear physics calculations beyond the threshold energies and with wavefunctions obtained by solving the Schr\"odinger equation with one--bosonexchange potentials. The constructed currents obey the nuclear form of thepartial conservation of the axial current. We apply the space component ofthese currents in calculations of the cross sections for the disintegration ofdeuterons by low energy (anti)neutrinos. The deuteron and the final statenucleon--nucleon wave functions are derived (i) from a variant of the OBEPQBpotential, and (ii) from the Nijmegen 93 and Nijmegen I nucleon-nucleoninteraction. The extracted values of the constant $L_{1, A}$, entering theaxial exchange currents of the pionless effective field theory, are in areasonable agreement with its value predicted by the dimensional analysis.
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