  We address the Noncommutative Noether's Problem on the invariants of Weyl
fields for linear actions of finite groups. We prove that if the variety
An(k)/G is rational then the Noncommutative Noether's Problem is positively
solved for G and any field k of characteristic zero. In particular, this gives
positive solution for all pseudo-reflections groups, for the alternating groups
(n = 3,4,5) and for any finite group when n = 3. Alternative proofs are given
for the complex field and for all pseudo-reflections groups. In the later case
an effective algorithm of finding the Weyl generators is described. We also
study birational equivalence for the rings of invariant differential operators
on complex affine irreducible varieties.
