  Currently, theory of ray transforms of vector and tensor fields is well
developed, but the Radon transforms of such fields have not been fully
analyzed. We thus consider linearly weighted and unweighted longitudinal and
transversal Radon transforms of vector fields. As usual, we use the standard
Helmholtz decomposition of smooth and fast decreasing vector fields over the
whole space. We show that such a decomposition produces potential and
solenoidal components decreasing at infinity fast enough to guarantee the
existence of the unweighted longitudinal and transversal Radon transforms of
these components.
  It is known that reconstruction of an arbitrary vector field from only
longitudinal or only transversal transforms is impossible. However, for the
cases when both linearly weighted and unweighted transforms of either one of
the types are known, we derive explicit inversion formulas for the full
reconstruction of the field. Our interest in the inversion of such transforms
stems from a certain inverse problem arising in magnetoacoustoelectric
tomography (MAET). The connection between the weighted Radon transforms and
MAET is exhibited in the paper. Finally, we demonstrate performance and noise
sensitivity of the new inversion formulas in numerical simulations.
