  We consider the antiferromagnetic Heisenberg model and the repulsive Hubbard
model for a class of frustrated lattices with a completely dispersionless
(flat) lowest one-particle (either one-magnon or one-electron) band. We
construct exact many-particle ground states for a wide range of particle
densities, calculate their degeneracy, and, as a result, obtain closed-form
expressions for the low-temperature thermodynamic quantities around a
particular value of the magnetic field $h_{\rm{sat}}$ or the chemical potential
$\mu_0$. We confirm our analytic findings by numerical data for finite
lattices.
