  We consider tight-binding single particle lattice Hamiltonians which are
invariant under an antiunitary antisymmetry: the anti-$\mathcal{PT}$ symmetry.
The Hermitian Hamiltonians are defined on $d$-dimensional non-Bravais lattices.
For an odd number of sublattices, the anti-$\mathcal{PT}$ symmetry protects a
flatband at energy $E = 0$. We derive the anti-$\mathcal{PT}$ constraints on
the Hamiltonian and use them to generate examples of generalized kagome
networks in two and three lattice dimensions. Furthermore, we show that the
anti-$\mathcal{PT}$ symmetry persists in the presence of uniform DC fields and
ensures the presence of flatbands in the corresponding irreducible
Wannier-Stark band structure. We provide examples of the Wannier-Stark band
structure of generalized kagome networks in the presence of DC fields, and
their implementation using Floquet engineering.
