Euler solutions of pseudodifferential equations

We consider a homogeneous pseudodifferential equation on a cylinder C = R x X over a smooth compact closed manifold X whose symbol extends to a meromo rphic function on the complex plane with values in the algebra of pseudodif ferential operators over X. When assuming the symbol to be independent on t he variable t is an element of R, we show an explicit formula for solutions of the equation. Namely, to each non-bijectivity paint of the symbol in th e complex plane there corresponds a finite-dimensional space of solutions, every solution being the residue of a meromorphic form manufactured from th e inverse symbol. In particular, for differential equations we recover Eule r's theorem on the exponential solutions. Our setting is model far the anal ysis on manifolds with conical points since C can be thought of as a 'stret ched' manifold with conical points at t = -infinity and t = infinity, When compared with the general theory, our approach is constructive while highli ghting all the features of this latter.