The roots of a smooth curve of hyperbolic polynomials may not in general be parameterized smoothly, even not C1,a for any a > 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 C3n 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.