For four types of functions ξ : ]0, ∞[→]0, ∞[, we characterize the law of two independent and positive r.v.'s X and Y such that U := ξ(X + Y) and V := ξ(X) − ξ(X + Y) are independent. The case ξ(x) = 1/x has been treated by Letac and Weso lowski (2000). As for the three other cases, under the weak assumption that X and Y have density functions whose logarithm is locally integrable, we prove that the distribution of (X, Y) is unique. This leads to Kummer, gamma and beta distributions. This improves the result obtained in [1] where more regularity was required from the densities.