A scaling system  is created when the potential obeys
To summarize, in dimensions in a scaling potential (of which the billiard is a special case), counting the `special' degrees of freedom gives: for translations (vector ), for rotations (antisymmetric part of ), and 1 for dilation (isotropic part of ). The total is .
I have strong numerical evidence that dilation is the only new special deformation which always arises when a hard-walled limit is taken of a general potential. Certainly the above arguments are sufficient to exclude simple cases, such as shear-type deformations. However I cannot exclude the possibility that a which is not differentiable everywhere in space allows new special functions to arise, which are not expressible as the Taylor series of (D.6). Also worthy of study is the general Hamiltonian system , no longer restricted to a constant mass tensor . This restriction played a key role in the above arguments.