  We consider an analog of the problem Veblen formulated in 1928 at the IMC:
classify invariant differential operators between "natural objects" (spaces of
either tensor fields, or jets, in modern terms) over a real manifold of any
dimension. For unary operators, the problem was solved by Rudakov (no nonscalar
operators except the exterior differential); for binary ones, by Grozman (there
are no operators of orders higher than 3, operators of order 2 and 3 are, bar
an exception in dimension 1, compositions of order 1 operators which, up to
dualization and permutation of arguments, form 8 families). In dimension one,
Grozman discovered an indecomposable selfdual operator of order 3 that does not
exist in higher dimensions. We solve Veblen's problem in the 1-dimensional case
over the ground field of positive characteristic. In addition to analogs of the
Berezin integral (strangely overlooked so far) and binary operators constructed
from them, we discovered two more (up to dualization) types of indecomposable
operators of however high order: analogs of the Grozman operator and a
completely new type of operators.
