  I provide an explicit construction of spectral curves for the affine
$\mathrm{E}_8$ relativistic Toda chain. Their closed form expression is
obtained by determining the full set of character relations in the
representation ring of $\mathrm{E}_8$ for the exterior algebra of the adjoint
representation; this is in turn employed to provide an explicit construction of
both integrals of motion and the action-angle map for the resulting integrable
system.
  I consider two main areas of applications of these constructions. On the one
hand, I consider the resulting family of spectral curves in the context of the
correspondences between Toda systems, 5d Seiberg-Witten theory, Gromov-Witten
theory of orbifolds of the resolved conifold, and Chern-Simons theory to
establish a version of the B-model Gopakumar-Vafa correspondence for the
$\mathrm{sl}_N$ L\^e-Murakami-Ohtsuki invariant of the Poincar\'e integral
homology sphere to all orders in $1/N$. On the other, I consider a degenerate
version of the spectral curves and prove a 1-dimensional Landau-Ginzburg mirror
theorem for the Frobenius manifold structure on the space of orbits of the
extended affine Weyl group of type $\mathrm{E}_8$ introduced by Dubrovin-Zhang
(equivalently, the orbifold quantum cohomology of the type-$\mathrm{E}_8$
polynomial $\mathbb{C} P^1$ orbifold). This leads to closed-form expressions
for the flat co-ordinates of the Saito metric, the prepotential, and a higher
genus mirror theorem based on the Chekhov-Eynard-Orantin recursion. I will also
show how the constructions of the paper lead to a generalisation of a
conjecture of Norbury-Scott to ADE $\mathbb{P}^1$-orbifolds, and a mirror of
the Dubrovin-Zhang construction for all Weyl groups and choices of marked
roots.
