  This paper concerns the time growth of the highest-order energy of the
systems of incompressible isotropic elastodynamics in two space dimensions. The
global well-posedness of smooth solutions near equilibrium was first obtained
by Lei [31] where the highest-order generalized energy may have certain growth
in time. We improve above result and show that the highest-order generalized
energy is uniformly bounded for all the time. The two dimensional
incompressible elastodynamics is a system of nonlocal quasilinear wave
equations where the unknowns decay as $\langle t\rangle^{-\frac12}$. This
suggests the problem is supercritical in the sense that the decay rate is far
from integrable. Surprisingly, we showed that in the highest-order energy
estimate, the temporal decay can be strongly enhanced to be subcritical
$\langle t \rangle^{-\frac54}$. The analysis is based on the ghost weight
energy method by Alinhac [5], the inherent strong null structure by Lei [31]
and the inherent div-curl structure of the system.
