  We prove that if $X,Y$ are positive, independent, non-Dirac random variables
and if for $\alpha,\beta\ge 0$, $\alpha\neq \beta$, $$
\psi_{\alpha,\beta}(x,y)=\left(y\,\tfrac{1+\beta(x+y)}{1+\alpha x+\beta
y},\;x\,\tfrac{1+\alpha(x+y)}{1+\alpha x+\beta y}\right), $$ 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 $$
U=\tfrac{Y}{1+X}\quad\mbox{and}\quad V= X\left(1+\tfrac{Y}{1+X}\right), $$
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.
