 The density "P"("x", "y") defines the invariant measure for this transformation.It is natural to suppose that "P"("x", "y") is a symmetric function of "x" and "y".This means that the measure is invariant with respect to the transformation "S"("x", "y") = ("y", "x") and hence with respect to the product "ST" with "ST"("x", "y") = ("x″", "y″") and

formula_99

Evidently "ST" has a first integral "H" = 1/"x" + "y".