  We study operators on a singular manifold, here of conical or edge type, and
develop a new general approach of representing asymptotics of solutions to
elliptic equations close to the singularities. The idea is to construct
so-called asymptotic parametrices with flat left -over terms. Our structures
are motivated by models of particle physics with singular potentials that
contribute embedded singularities in $\R^N$ of higher order, according to the
number of particles.
