  This paper studies the behavior under iteration of the maps
T_{ab}(x,y)=(F_{ab}(x)-y,x) of the plane R^2, in which F_{ab}(x)=ax if x>=0 and
bx if x<0. The orbits under iteration correspond to solutions of the nonlinear
difference equation x_{n+2}= 1/2(a-b)|x_{n+1}| + 1/2(a+b)x_{n+1} - x_n. This
family of maps has the parameter space (a,b)\in R^2. These maps are
area-preserving homeomorphisms of R^s that map rays from the origin into rays
from the origin. This paper shows the existence of special parameter values
where T_{ab} has every nonzero orbit an invariant circle with irrational
rotation number, and these invariant circles are piecewise unions of arcs of
conic sections. Numerical experiments suggest the possible existence of many
other parameter values having invariant circles.
