  A bounded domain $K \subset \mathbb R^n$ is called polynomially integrable if
the $(n-1)$-dimensional volume of the intersection $K$ with a hyperplane $\Pi$
polynomially depends on the distance from $\Pi$ to the origin. It was proved in
[7] that there are no such domains with smooth boundary if $n$ is even, and if
$n$ is odd then the only polynomially integrable domains with smooth boundary
are ellipsoids. In this article, we modify the notion of polynomial
integrability for even $n$ and consider bodies for which the sectional volume
function is a polynomial up to a factor which is the square root of a quadratic
polynomial, or, equivalently, the Hilbert transform of this function is a
polynomial. We prove that ellipsoids in even dimensions are the only convex
infinitely smooth bodies satisfying this property.
