Generators and defining relations for ring of invariants of commuting locally nilpotent derivations or automorphisms

Let A be an algebra over a field K of characteristic zero and let E DerK(A) be commuting locally nilpotent K-derivations such that delta(i) (x(j)) equals delta(ij), the Kronecker delta, for some elements x(1), ..., x, epsilon A. A set of generators for the algebra A(delta) := boolean AND(s)(i)=1 ker(delta(i)) is found explicitly and a set of defining relations for the algebra A(delta) is described. Similarly, let sigma(1),..., sigma(s), epsilon Aut(K) (A) be commuting K-automorphisms of the algebra A is given such that the maps sigma(i) - id(A) are locally nilpotent and sigma(i)(x(j)) = x(j) + delta(ij), for some elements x(1),..., x. E A. A set of generators for the algebra A(sigma) : = {a epsilon A vertical bar sigma(1) (a) =... = sigma(s) (a) = a} is found explicitly and a set of defining relations for the algebra A(delta) is described. In general, even for a finitely generated non-commutative algebra A the algebras of invariants A(delta) and A(sigma) are not finitely generated, not (left or right) Noetherian and a minimal number of defining relations is infinite. However, for a finitely generated commutative algebra A the opposite is always true. The derivations (or automorphisms) just described appear often in many different situations (possibly) after localization of the algebra A.