A RADON TYPE TRANSFORM RELATED TO THE EULER EQUATIONS FOR IDEAL FLUID

We study the Nadirashvili - Vladuts transform N that integrates second rank tensor fields f on Rn over hyperplanes. More precisely, for a hyperplane P and vector. parallel to P, Nf(P,.) is the integral of the function fij(x). i.j over P, where. is the unit normal vector to P. We prove that, given a vector field v, the tensor field f = v. v belongs to the kernel of N if and only if there exists a function p such that (v, p) is a solution to the Euler equations. Then we study the Nadirashvili - Vladuts potential w(x, xi) determined by a solution to the Euler equations. The function w solves some 4th order PDE. We describe all solutions to the latter equation.