  Let $T_1,...,T_n$ denote free random variables. For two linear forms
$L_1=\sum_{j=1}^n a_jT_j$ and $L_2=\sum_{j=1}^n b_jT_j$ with real coefficients
$a_j$ and $b_j$ we shall describe all distributions of $T_1,...,T_n$ such that
$L_1$ and $L_2$ are free. For identically distributed free random variables
$T_1,...,T_n$ with distribution $\mu$ we establish necessary and sufficient
conditions on the coefficients $a_j,b_j,\,j=1,...,n,$ such that the
statements:\quad $(i)$ $\mu$ is a centered semicircular distribution; and
$(ii)$ \, $L_1$ and $L_2$ are identically distributed
($L_1\stackrel{D}{=}L_2$); are equivalent. In the proof we give a complete
characterization of all sequences of free cumulants of measures with compact
support and with a finite number of non zero entries. The characterization is
based on topological properties of regions defined by means of the Voiculescu
transform $\phi$ of such sequences.
