 On the line "H" = const ≡ "c" the transformation has the form

formula_100

Hence the invariant measure density of "ST" must be of the form

formula_101

Taking account of the symmetry "P"("x", "y")= "P"("y", "x") this becomes "f"("c")= "c"−2 and hence (after normalization)
(its integration over "x"+ or "x"– yields the function "w"("x") ).The reduction of the transformation to one-to-one mapping was used already by  and they obtained a formula of the form of but for other variables; their paper does not contain applications to the problems which are considered in.The correctness of be verified also by a direct calculation; the Jacobian of the transformation is

formula_102

(in its calculation one must note that formula_103).