q-Difference equations of KdV type and "Chazy-type" second-degree difference equations

By imposing special compatible similarity constraints on a class of integrable partial $q$-difference equations of KdV-type we derive a hierarchy of second-degree ordinary $q$-difference equations. The lowest (non-trivial) member of this hierarchy is a second-order second-degree equation which can be considered as an analogue of equations in the class studied by Chazy. We present corresponding isomonodromic deformation problems and discuss the relation between this class of difference equations and other equations of Painleve type.