In order to remove the collision singularity in the equations of motion of the three dimensional two body problem, one defines the coordinate transformation
$x_1=u_1^2-u_2^2-u_3^2+u_4^2$
$x_2=2(u_1 u_2-u_3 u_4)$
$x_3=2(u_1 u_3+u_2 u_4)$
Known as the K-S transformation, certainly this doesn't preserve the dimension. Does anyone know if another transformation can be defined such that preserve the dimension and remove the collision singularity? Thanks and i apologize for my english in advance.