  Consider the nonlinear and completely continuous scattering map \[
\mathcal{S}\big((\Omega; \lambda, \mu, V),
\mathbf{u}^i\big)=\mathbf{u}_t^\infty(\hat{\mathbf{x}}), \quad
\hat{\mathbf{x}}\in\mathbb{S}^{n-1}, \] which sends an inhomogeneous elastic
scatterer $(\Omega; \lambda, \mu, V)$ to its far-field pattern
$\mathbf{u}_t^\infty$ due to an incident wave field $\mathbf{u}^i$ via the
Lam\'e system. Here, $(\lambda, \mu, V)$ signifies the medium configuration of
an elastic scatterer that is compactly supported in $\Omega$. In this paper, we
are concerned with the intrinsic geometric structure of the kernel space of
$\mathcal{S}$, which is of fundamental importance to the theory of inverse
scattering and invisibility cloaking for elastic waves and has received
considerable attention recently. It turns out that the study is contained in
analysing the geometric properties of a certain non-selfadjoint and
non-elliptic transmission eigenvalue problem. We propose a generalized elastic
transmission eigenvalue problem and prove that the transmission eigenfunctions
vanish locally around a corner of $\partial\Omega$ under generic regularity
criteria. The regularity criteria are characerized by the H\"older continuity
or a certain Fourier extension property of the transmission eigenfunctions. As
an interesting and significant application, we apply the local geometric
property to derive several novel unique identifiability results for a
longstanding inverse elastic problem by a single far-field measurement.
