We study curves of fixed points for certain diffeomorphisms of R3 that are induced by automorphisms of a trace algebra. We classify these curves. There is a function E which is invariant under all such trace maps and the level surfaces Et:E=t are invariant; a point of Et will be said to have level t. The surface E1 is significant. Then most fixed points on E1 are actually on a curve γ of fixed points interior to E1. We describe the possibilities for the other end of γ on E1