Hidden Poisson structures related to Hill's equation
As argued by George Wilson, the Miura map $v\mapsto u$ may be seen as part of a chain of field extensions
$\CC\!<\!u\!>\subset\CC\!<\!v\!>\subset\CC\!<\!\phi,\psi\!>$. By (i) incorporating monodromy in the space of solutions to
the linear problem $\psi''+u\psi=0$ and
(ii) extending the notion of Poisson symmetry to that of
Poisson Lie group type, the result of Wilson is shown to be
an example of Poisson reduction. The result is extended to
the lattice analogue by direct means.