Smooth counterexamples to strong unique continuation for a Beltrami
system in C^2

Adam Coffman, Yifei Pan

article abstract

We construct an example of a smooth map C to C^2 which vanishes to
infinite order at the origin, and such that the ratio of the norm of
the z-bar derivative to the norm of the z derivative also vanishes to
infinite order.  This gives a counterexample to strong unique
continuation for a vector valued analogue of the Beltrami equation.