Rectification of spheres of co-dimension 1 and 2 in Rn

Consider a pencil S of k-dimensional surfaces in Rn passing through the origin. A rectification of S is a germ Φ of a diffeomorphism (Rn,0)→(Rn,0) such that the image Φ(s) of each surface s∈S belongs to an affine k-subspace. Here Φ(s) denotes, more precisely, the restriction of Φ to a germ of such a surface s. The main result of the paper is the following. Let S be a rectifiable pencil of spheres in Rn of co-dimension 1 or 2. Assume that S is large enough and that the tangent planes to spheres in S are in general position. Then all spheres in S have a common point different from the origin.