Mappings that preserve cones in Lobachevskii space

Let $\textЛ^n$ be $n$-dimensional Lobachevskii space, and $\{l_x:X\in\textЛ^n\}$ be a family of lines, parallel to a line $l_o$, $o\in\textЛ^n$ (in a given direction). Let $\{C_x:X\in\textЛ^n\}$ be a family of circular cones in $\textЛ^n$ of opening $\alpha$ with axes $l_X$ and vertex $X$. Then, if $f:\textЛ^n\to\textЛ^n$ ($n>2$) is a bijective mapping and $f(Cx)=C_{f(x)}$, it follows thatf is a motion in the space $\textЛ^n$.