Handling ray transform of symmetric tensor fields and the Radon transform of differential forms on incomplete data

A symmetric tensor field in the Euclidian space of any dimension ${{\mathbb R}}^n$ can be reconstructed from its integrals along straight lines which form n-manifold. We obtained the new formulas reconstructing solenoidal part of the fields when a ray transform is known only on complex lines, intersecting a given curve, as well as a curve belonging to infinity. The new family of two-dimensional planes in ${{\mathbb R}}^n$ is introduced and the inversion formula is obtained for the case of differential forms of degree $2$ on the given integrals along the planes of this family.