  We extend the result of D. Phillips (On one-homogeneous solutions to elliptic
systems in two dimensions. C. R. Math. Acad. Sci. Paris 335 (2002), no. 1,
39-42) by showing that one-homogeneous solutions of certain elliptic systems in
divergence form either do not exist or must be affine. The result is novel in
two ways. Firstly, the system is allowed to depend (in a sufficiently smooth
way) on the spatial variable x. Secondly, Phillips's original result is shown
to apply to one-homogeneous solutions belonging to the Sobolev space H^{1},
from which his treatment of Lipschitz solutions follows as a special case. A
singular one-homogeneous solution to an elliptic system violating the
hypotheses of the main theorem is constructed using a variational method which
has links to nonlinear elasticity.
