  In our earlier paper [9], it is proved that a homogeneous rigid, traction or
impedance condition on one or two intersecting line segments together with a
certain zero point-value condition implies that the solution to the Lam\'e
system must be identically zero, which is referred to as the generalized
Holmgren principle (GHP). The GHP enables us to solve a longstanding inverse
scattering problem of determining a polygonal elastic obstacle of general
impedance type by at most a few far-field measurements. In this paper, we
include all the possible physical boundary conditions from linear elasticity
into the GHP study with additionally the soft-clamped, simply-supported as well
as the associated impedance-type conditions. We derive a comprehensive and
complete characterisation of the GHP associated with all of the aforementioned
physical conditions. As significant applications, we establish novel unique
identifiability results by at most a few scattering measurements not only for
the inverse elastic obstacle problem but also for the inverse elastic
diffraction grating problem within polygonal geometry in the most general
physical scenario. We follow the general strategy from [9] in establishing the
results. However, we develop technically new ingredients to tackle the more
general and challenging physical and mathematical setups. It is particularly
worth noting that in [9], the impedance parameters were assumed to be constant
whereas in this work they can be variable functions.
