  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.
