  Wigner integrals and the corresponding Wigner chaos were introduced by P.
Biane and R. Speicher in 1998 as a non-commutative counterpart of classical
Wiener-It\^o integrals and the corresponding Wiener-It\^o chaos, respectively,
in free probability.
  In the classical case, a famous result of I. Shigekawa states that
non-trivial elements in the finite Wiener-It\^o chaos have an absolutely
continuous distribution. We provide here a first contribution to such
regularity questions for Wigner integrals by showing that the distribution of
non-trivial elements in the finite Wigner chaos cannot have atoms. This answers
a question of I. Nourdin and G. Peccati.
  For doing so, we establish the notion of directional gradients in the context
of the free Malliavin calculus. These directional gradients bridge between free
Malliavin calculus and the theory of non-commutative derivations as initiated
by D. Voiculescu and Y. Dabrowski. Methods recently invented by R. Speicher, M.
Weber, and the author for treating similar questions in the case of finitely
many variables are extended, such that they apply to directional gradients.
This approach also excludes zero-divisors for the considered elements in the
finite Wigner chaos.
