  An $SU({\rm n})$-symmetric generalization of the Heisenberg model for quantum
spin $S$ operators is used to investigate the geometrically frustrated
body-centered tetragonal (BCT) lattice with antiferromagnetic interlayer
coupling $J_1$ and intralayer first and second neighbor coupling $J_2$ and
$J_3$. Using complementary representations of the spin operators, we study the
phase diagram characterizing the ground state of the system. For small n, we
find that the most stable solutions correspond to four different families of
possible long range magnetic orders that are governed by $J_1$, $J_2$, and
$J_3$. First, some possible instabilities of these phases are identified for
$n=2$ in large $S$ expansions up to the linear spin-wave corrections. Then,
using a fermionic representation of the $SU({\rm n})$ spin operators for
$S=1/2$, we find that purely magnetic orders occur for ${\rm n}\le 3$ while
spin-liquid (SL) solutions are stabilized for ${\rm n}\ge 10$. The SL solution
governed by $J_1$ breaks the lattice translation symmetry. This Modulated SL is
associated to a commensurate ordering wave vector $(1,1,1)$. For $4\le {\rm
n}\le 9$, we show how competition between $J_1$, $J_2$, and $J_3$ can tune the
ground state from beeing magnetically ordered to a SL state. We discuss the
relevance of this scenario for correlated systems with BCT crystal structure.
