  We classify real hypersurfaces with isometric Reeb flow in the complex
quadrics Q^m for m > 2. We show that m is even, say m = 2k, and any such
hypersurface is an open part of a tube around a k-dimensional complex
projective space CP^k which is embedded canonically in Q^{2k} as a totally
geodesic complex submanifold. As a consequence we get the non-existence of real
hypersurfaces with isometric Reeb flow in odd-dimensional complex quadrics.
