  We study here properties of free Generalized Inverse Gaussian distributions
(fGIG) in free probability. We show that in many cases the fGIG shares similar
properties with the classical GIG distribution. In particular we prove that
fGIG is freely infinitely divisible, free regular and unimodal, and moreover we
determine which distributions in this class are freely selfdecomposable. In the
second part of the paper we prove that for free random variables $X,Y$ where
$Y$ has a free Poisson distribution one has $X\stackrel{d}{=}\frac{1}{X+Y}$ if
and only if $X$ has fGIG distribution for special choice of parameters. We also
point out that the free GIG distribution maximizes the same free entropy
functional as the classical GIG does for the classical entropy.
