Title: Miura transformations for discrete Painlevé equations coming from the affine E$_8$ Weyl group,
Abstract: We derive integrable equations starting from autonomous mappings with a
general form inspired by the additive systems associated to the affine Weyl
group E$_8^{(1)}$. By deautonomisation we obtain two hitherto unknown systems,
one of which turns out to be a linearisable one, and we show that both these
systems arise from the deautonomisation of a non-QRT mapping. In order to
unambiguously prove the integrability of these nonautonomous systems, we
introduce a series of Miura transformations which allows us to prove that one
of these systems is indeed a discrete Painlevé equation, related to the
affine Weyl group E$_7^{(1)}$, and to cast it in canonical form. A similar
sequence of Miura transformations allows us to effectively linearise the second
system we obtain. An interesting off-shoot of our calculations is that the
series of Miura transformations, when applied at the autonomous limit, allows
one to transform a non-QRT invariant into a QRT one.