Huygens' principle and separation of variables

We demonstrate a close relation between the algebraic structure of the (local) group of conformal transformations on a smooth Lorentzian manifold M and the existence of nontrivial hierarchies of wave-type hyperbolic operators satisfying Huygens' principle on M. The mechanism of such a relation is provided through a local separation of variables for linear second order partial differential operators with a metric principal symbol. The case of fat (Minkowski) spaces is studied in detail. As a result, some new nontrivial classes of Huygens operators are constructed. Their relation to the classical Hadamard conjecture and its modifications is discussed.