  We study the emission of neutrinos, resulting from the scattering of
electrons off magnetic flux tubes (fluxoids) in the neutron star cores with
superfluid (superconducting) protons. In the absence of proton superfluidity
(T> T_{cp}), this process transforms into the well known electron synchrotron
emission of neutrino pairs in a locally uniform magnetic field B, with the
neutrino energy loss rate Q proportional to B^2 T^5. For temperatures T not
much below T_{cp}, the synchrotron regime (Q \propto T^5) persists and the
emissivity Q can be amplified by several orders of magnitude due to the
appearance of the fluxoids and associated enhancement of the field within them.
For lower T, the synchrotron regime transforms into the bremsstrahlung regime
(Q \propto T^6) similar to the ordinary neutrino-pair bremsstrahlung of
electrons which scatter off atomic nuclei. We calculate Q numerically and
represent our results through a suitable analytic fit. In addition, we estimate
the emissivities of two other neutrino-production mechanisms which are usually
neglected -- neutrino-pair bremsstrahlung processes due to electron-proton and
electron-electron collisions. We show that the electron-fluxoid and
electron-electron scattering can provide the main neutrino production
mechanisms in the neutron star cores with highly superfluid protons and
neutrons at T < 5 10^8 K. The electron-fluxoid scattering is significant if the
initial, locally uniform magnetic field B > 10^{13} G.
