Let B=k[x,y,z,u]$B=k[x,y,z,u]$ be a polynomial ring in four variables over a field k$k$ of characteristic 0$0$, and let X$X$ be 4-dimensional affine space over k$k$. A locally nilpotent derivation D$D$ of B$B$ induces an algebraic action of the additive group G=(k,+)$G=(k,+)$ on X$X$ via the exponential mapping exp (tD)${\rm exp} (tD)$, t in k$t \in k$. The ring of invariants of the group action is equal to the kernel of the derivation, which is a UFD of transcendence degree 3$3$ over k. $k$ (It is not known whether the kernel must be affine.).
Consider the triangular derivation D: u--> z--> y--> x^n$D: u \to z \to y \to x^n$ and x-->0$x \to 0$, where n$n$ is a positive integer. The kernel of D$D$ is of the form k[x,f,g,h]$k[x,f,g,h]$, where x^(2n)u+f^3+g^2=0$x^{2n}u+f^3+g^2=0$. The corresponding algebraic variety X/G$X/G$ is therefore of the type you are looking for.
