  Hermitian bipartite models are characterized by the presence of chiral
symmetry and by Lieb's theorem, which derives the number of zero-energy flat
bands of the model from the imbalance of sites between its two sublattices.
Here, we introduce a class of non-Hermitian models with an arbitrary number of
sublattices connected in a unidirectional and cyclical way and show that the
number of zero-energy flat bands of these models can be found from a
generalized version of Lieb's theorem, in what regards its application to
noninteracting tight-binding models, involving the imbalance between each
sublattice and the sublattice of lowest dimension. Furthermore, these models
are also shown to obey a generalized chiral symmetry, of the type found in the
context of certain clock or parafermionic systems. The main results are
illustrated with a simple toy model, and possible realizations in different
platforms of the models introduced here are discussed.
