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, let $\d_1,

…, \d_s\in \Der_K(A)$be\text{em}commutinglocallynilpotent$K$-derivationssuchthat$\d_i(x_j)=\d_{ij}$, the Kronecker delta, for some elements $x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_s\in A$. A set of algebra generators for the algebra $A^{\text{d}}:={\cap }_{i=1}^s\ker ({\text{d}}_i)$ is found {\em explicitly} and a set of {\em defining relations} for the algebra $A^{\text{d}}$ is described. Similarly, given a set ${\text{s}}_1,\mathrm{.}\mathrm{.}\mathrm{.},{\text{s}}_s\in {\text{Aut}}_K(A)$ of {\em commuting} $K$-automorphisms of the algebra $A$ such that the maps ${\text{s}}_i-\mathrm{i}{\mathrm{d}}_{\mathrm{A}}$ are {\em locally nilpotent} and ${\text{s}}_i(x_j)=x_j+{\text{d}}_{ij}$, for some elements $x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_s\in A$. A set of algebra generators for the algebra $A^\s:={a\in A | \s_1(a)=… =\s_s(a)=a}$ is found {\em explicitly} and a set of defining relations for the algebra $A^{\text{s}}$ is described. In general, even for a {\em finitely generated noncommutative} algebra $A$ the algebras of invariants $A^{\text{d}}$ and $A^{\text{s}}$ are {\em not} finitely generated, {\em not} (left or right) Noetherian and {\em does not} satisfy finitely many defining relations (see examples). Though, for a {\em finitely generated commutative} algebra $A$ {\em always} the {\em opposite} is true. The derivations (or automorphisms) just described appear often in may different situations after (possibly) a localization of the algebra $A$.