  Flat bands are of extreme interest in a broad spectrum of fields since given
by their high degeneracy, a small perturbation introduced in the system is able
to push the ground state in the direction of an ordered phase of interest.
Hence the flat band engineering in real materials attracts huge attention.
However, manufacturing a flat band represents a difficult task because its
appearance in a real system is connected to rigid mathematical conditions
relating a part of Hamiltonian parameters. Consequently, whenever a flat band
is desired to be manufactured, these Hamiltonian parameters must be tuned
exactly to the values fixed by these rigid mathematical conditions. Here we
demonstrate that taking the many-body spin-orbit interaction (SOI) into account
-- which can be continuously tuned e.g. by external electric fields --, these
rigid mathematical conditions can be substantially relaxed. On this line we
show that a $\sim 20-30 \%$ variation in the Hamiltonian parameters rigidly
fixed by the flat band conditions can also lead to flat bands in the same or in
a bit displaced position on the energy axis. This percentage can even increase
to $\sim 80 \%$ in the presence of an external magnetic field. The study is
made in the case of conducting polymers. These systems are relevant not only
because they have broad application possibilities, but also because they can be
used to present the mathematical background of the flat band conditions in full
generality, in a concise, clear and understandable manner applicable everywhere
in itinerant systems.
