Constants of derivations in polynomial rings over unique factorization domains

A well-known theorem, due to Nagata and Nowicki, states that the ring of constants of any ${\mathcal K}$-derivation of ${\mathcal K}[x,y]$, where ${\mathcal K}$ is a commutative field of characteristic zero, is a polynomial ring in one variable over ${\mathcal K}$. In this paper we give an elementary proof of this theorem and show that it remains true if we replace ${\mathcal K}$ by any unique factorization domain of characteristic zero.