  We present in detail the Einstein equations in the
Baumgarte-Shapiro-Shibata-Nakamura formulation for the case of $D$ dimensional
spacetimes with $SO(D-d)$ isometry based on a method originally introduced in
Ref.1. Regularized expressions are given for a numerical implementation of this
method on a vertex centered grid including the origin of the quasi-radial
coordinate that covers the extra dimensions with rotational symmetry.
Axisymmetry, corresponding to the value $d=D-2$, represents a special case with
fewer constraints on the vanishing of tensor components and is conveniently
implemented in a variation of the general method. The robustness of the scheme
is demonstrated for the case of a black-hole head-on collision in $D=7$
spacetime dimensions with $SO(4)$ symmetry.
