Transversals in splitted Latin squares of even order

We consider the splitted Latin squares, i.e. Latin squares of order $2n$ with elements from $\{0,\ldots,2n-1\}$ such that after reducing modulo $n$ we obtain $2n\times2n$-matrix consisting of four Latin squares of order $n$. The set of all transversals of splitted Latin square is described by means of $2$-balansed multisets of entries of one of Latin squares of order $n$ mentioned above. A quick algorithm of construction (after some preliminary work) the set of all transversals for any splitted Latin square of order $2n$ corresponding to an arbitrary set of four Latin squares of order $n$ is described.