problem.]{} [*Assume that $H \subseteq G$ is a closed inclusion of affine algebraic groups, and that $A$ is a finitely generated commutative $\Bbbk$–algebra. Assume that $G$ acts rationally in the affine algebra $A$. Find conditions for the pair $(H,G)$ that guarantee that if $A^G$ is finitely generated so is $A^H$.*]{}

It is clear that Theorem \[theo:descent\] guarantees that if $G$ is reductive then, the generalized Hilbert’s $14^{\text{th}}$ problem has a positive answer if $H$ is observable in $G$.

The perspective of observability in Hilbert’s $\boldsymbol{14}^{\text{th}}$ problem
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The original formulation by D. Hilbert of his famous $14^{\text{th}}$ problem reads as follows (as it appeared translated into English in [@kn:hilbertpr]):

> “By a finite field of integrality I mean a system of functions from which a finite number of functions can be chosen, in which all other functions of the system are rationally and integrally expressible. Our problem amounts to this: to show that all relatively integral functions of any given domain of rationality always constitute a finite field of integrality”.

In modern language this problem can be formulated as follows –see [@kn:Mumsur]–: “Let $\Bbbk $ be a field \[$ \{x_{1},\dots,x_{n}\}$ a family of indeterminates\] and let $K$ be a subfield 