Isometries between spaces of vector-valued functions of bounded variation

Let $X$ and $Y$ be subsets of the real line with at least two points, and let $E$ and $F$ be strictly normed spaces. By $BV(X,E)$ we mean the space of $E$-valued functions of bounded variation on $X$ with respect to the supremum norm $\|\cdot\|_\infty$. In this paper we give a complete description of a surjective, not assumed to be linear, isometry $T: BV(X,E) \longrightarrow BV(Y,F)$.