  We consider flame front propagation in channel geometries. The steady state
solution in this problem is space dependent, and therefore the linear stability
analysis is described by a partial integro-differential equation with a space
dependent coefficient. Accordingly it involves complicated eigenfunctions. We
show that the analysis can be performed to required detail using a finite order
dynamical system in terms of the dynamics of singularities in the complex
plane, yielding detailed understanding of the physics of the eigenfunctions and
eigenvalues.
