Independence preserving property of Kummer laws

We prove that if $X,Y$ are positive, independent, non-Dirac random variables and if for $\alpha ,\beta \ge 0$, $\alpha \mathrm{\ne }\beta$, $${\psi }_{\alpha ,\beta }(x,y)=(y\frac{1+\beta (x+y)}{1+\alpha x+\beta y},x\frac{1+\alpha (x+y)}{1+\alpha x+\beta y}),$$ then the random variables $U$ and $V$ defined by $(U,V)={\psi }_{\alpha ,\beta }(X,Y)$ are independent if and only if $X$ and $Y$ follow Kummer distributions with suitably related parameters. In other words, any invariant measure for a lattice recursion model governed by ${\psi }_{\alpha ,\beta }$ in the scheme introduced by Croydon and Sasada in \cite{CS2020} is necessarily a product measure with Kummer marginals. The result extends earlier characterizations of Kummer and gamma laws by independence of $$U=\frac{Y}{1+X}\text{quadand}V=X(1+\frac{Y}{1+X}),$$ which corresponds to the case of ${\psi }_{1,0}$. We also show that this independence property of Kummer laws covers, as limiting cases, several independence models known in the literature: the Lukacs, the Kummer-Gamma, the Matsumoto-Yor and the discrete Korteweg de Vries models.