  Helmholtz decompositions of the elastic fields open up new avenues for the
solution of linear elastic scattering problems via boundary integral equations
(BIE) [Dong, Lai, Li, Mathematics of Computation,2021]. The main appeal of this
approach is that the ensuing systems of BIE feature only integral operators
associated with the Helmholtz equation. However, these BIE involve non standard
boundary integral operators that do not result after the application of either
the Dirichlet or the Neumann trace to Helmholtz single and double layer
potentials. Rather, the Helmholtz decomposition approach leads to BIE
formulations of elastic scattering problems with Neumann boundary conditions
that involve boundary traces of the Hessians of Helmholtz layer potential. As a
consequence, the classical combined field approach applied in the framework of
the Helmholtz decompositions leads to BIE formulations which, although robust,
are not of the second kind. Following the regularizing methodology introduced
in [Boubendir, Dominguez, Levadoux, Turc, SIAM Journal on Applied Mathematics
2015] we design and analyze novel robust Helmholtz decomposition BIE for the
solution of elastic scattering that are of the second kind in the case of
smooth scatterers in two dimensions. We present a variety of numerical results
based on Nystrom discretizations that illustrate the good performance of the
second kind regularized formulations in connections to iterative solvers.
