  We consider the problem of finding the spectrum of an operator taking the
form of a low-rank (rank one or two) non-normal perturbation of a
well-understood operator, motivated by a number of problems of applied interest
which take this form. We use the fact that the system is a low rank
perturbation of a solved problem, together with a simple idea of classical
differential geometry (the envelope of a family of curves) to completely
analyze the spectrum. We use these techniques to analyze three problems of this
form: a model of the oculomotor integrator due to Anastasio and Gad (2007), a
continuum integrator model, and a nonlocal model of phase separation due to
Rubinstein and Sternberg (1992).
