  The structure of diagonal singularities of Green functions of partial
differential operators of even order acting on smooth sections of a vector
bundle over a Riemannian man ifold is studied. A special class of operators
formed by the products of second-order operators of Laplace type defined with
the help of a unique Riemannian metric and a unique bundle connection but with
different potential terms is investigated. Explicit simple formulas for
singularities of Green functions of such operators in terms of the usual heat
kernel coefficients are obtained.
