  The roots of a smooth curve of hyperbolic polynomials may not in general be
parameterized smoothly, even not $C^{1,\alpha}$ for any $\alpha > 0$. A
sufficient condition for the existence of a smooth parameterization is that no
two of the increasingly ordered continuous roots meet of infinite order. We
give refined sufficient conditions for smooth solvability if the polynomials
have certain symmetries. In general a $C^{3n}$ curve of hyperbolic polynomials
of degree $n$ admits twice differentiable parameterizations of its roots. If
the polynomials have certain symmetries we are able to weaken the assumptions
in that statement.
