  In this work we use Lie group theoretic methods and the theory of prolonged
group actions to study two fully nonlinear partial differential equations
(PDEs). First we consider a third order PDE in two spatial dimensions that
arises as the analogue of the Euler-Lagrange equations from a second order
variational principle in $L^{\infty}$. The equation, known as the
$\infty$-Polylaplacian, is a higher order generalisation of the
$\infty$-Laplacian, also known as Aronsson's equation. In studying this problem
we consider a reduced equation whose relation to the $\infty$-Polylaplacian can
be considered analogous to the relationship of the Eikonal to Aronsson's
equation. Solutions of the reduced equation are also solutions of the
$\infty$-Polylaplacian. For the first time we study the Lie symmetries admitted
by these two problems and use them to characterise and construct invariant
solutions under the action of one dimensional symmetry subgroups.
