Discrete Painlev\'e equations and their Lax pairs as reductions of   integrable lattice equations
  We present a method of determining a Lax representation for similarity reductions of autonomous and non-autonomous partial difference equations. This method may be used to obtain Lax representations that are general enough to provide the Lax integrability for entire hierarchies of reductions. A main result is, as an example of this framework, how we may obtain the q-Painlev\'e equation whose group of B\"acklund transformations is an affine Weyl group of type E_6^{(1)} as a similarity reduction of the discrete Schwarzian Korteweg-de Vries equation. 
C. M. Ormerod, Peter H. van der Kamp and G. R. W. Quispel
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