  Let $k$ be a field of arbitrary characteristic. Nakai (1978) proved a
structure theorem for $k$-domains admitting a nontrivial locally finite
iterative higher derivation when $k$ is algebraically closed. In this paper, we
generalize Nakai's theorem to cover the case where $k$ is not algebraically
closed. As a consequence, we obtain a cancellation theorem of the following
form: Let $A$ and $A'$ be finitely generated $k$-domains with $A[x]\simeq
_kA'[x]$. If $A$ and $\bar{k}\otimes _kA$ are UFDs and $\mathop{\rm
trans.deg}\nolimits_kA=2$, then we have $A\simeq _kA'$. This generalizes the
cancellation theorem of Crachiola (2009).
