  The Painleve expansion for the second Painleve equation (PII) and fourth
Painleve equation (PIV) have two branches. The singular manifold method
therefore requires two singular manifolds. The double singular manifold method
is used to derive Miura transformations from PII and PIV to modified Painleve
type equations for which auto-Backlund transformations are obtained. These
auto-Backlund transformations can be used to obtain discrete equations.
