Title: On algebraically integrable domains in Euclidean spaces,
Abstract: 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?