Let $k$ be a field of characteristic $p \geq 0$ and $A = k[x_0, x_1, x_2, \ldots]$ the polynomial ring in countably many variables over $k$. We construct a rational higher $k$-derivation on $A$ whose kernel is not the kernel of any higher $k$-derivation on $A$. This example extends [5, Example 4].

Let $X$ be a log del Pezzo surface of rank one. In [8], the first author determined the possible singularity type of $X$ when $X$ contains the affine plane as a Zariski open subset. In this paper, we prove that, if $X$ contains a non-cyclic quotient singular point and its singularity type is one of the list of [8, Appendix C], then it contains the affine plane as a Zariski open subset.