  It is known that the Neumann--Poincar\'e operator for the Lam\'e system of
linear elasticity is polynomially compact and, as a consequence, that its
spectrum consists of three non-empty sequences of eigenvalues accumulating to
certain numbers determined by Lam\'e parameters, if the boundary of the domain
where the operator is defined is $C^\infty$-smooth. We extend this result to
less smooth boundaries, namely, $C^{1, \alpha}$-smooth boundaries for some
$\alpha > 0$. The results are obtained by proving certain identities for
surface Riesz transforms, which are singular integral operators of
nonconvolution type, defined by the matrix tensor on a given surface.
