  We consider the strongly anisotropic spin-1/2 $XXZ$ model on the
sawtooth-chain lattice with ferromagnetic longitudinal interaction
$J^{zz}=\Delta J$ and aniferromagnetic transversal interaction
$J^{xx}=J^{yy}=J>0$. At $\Delta=-1/2$ the lowest one-magnon excitation band is
dispersionless (flat) leading to a massively degenerate set of ground states.
Interestingly, this model admits a three-coloring representation of the
ground-state manifold [H.~J.~Changlani et at., Phys. Rev. Lett. {\bf 120},
117202 (2018)]. We characterize this ground-state manifold and elaborate the
low-temperature thermodynamics of the system. We illustrate the manifestation
of the flat-band physics of the anisotropic model by comparison with two
isotropic flat-band Heisenberg sawtooth chains. Our analytical consideration is
complemented by exact diagonalization and finite-temperature Lanczos method
calculations.
