The KLR-theorem revisited

For independent random variables $X_1,\dots ,X_n;Y_1,\dots ,Y_n$ with all $X_i$ identically distributed and same for $Y_j$, we study the relation $$E\{a\bar{X}+b\bar{Y}\mathrm{\mid }{X}_1-\bar{X}+Y_1-\bar{Y},\dots ,X_n-\bar{X}+Y_n-\bar{Y}\}=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$$ with $a,b$ some constants. It is proved that for $n\ge 3$ and $ab>0$ the relation holds iff $X_i$ and $Y_j$ are Gaussian.\ A new characterization arises in case of $a=1,b=-1$. In this case either $X_i$ or $Y_j$ or both have a Gaussian component. It is the first (at least known to the author) case when presence of a Gaussian component is a characteristic property.