On algebraically integrable domains in Euclidean spaces

Let $D$ be a bounded domain $D$ in ${\mathbb{R}}^n$ with infinitely smooth boundary and $n$ is odd. We prove that if the volume cut off from the domain by a hyperplane is an algebraic function of the hyperplane, free of real singular points, then the domain is an ellipsoid. This partially answers a question of V.I. Arnold: whether odd-dimensional ellipsoids are the only algebraically integrable domains?