We consider the Schrödinger operators on zigzag nanoribbons (tight-binding models) in external magnetic and electric fields. If these fields are absent, then the spectrum of the Schrödinger operator consists of two non-flat bands and one flat band (an eigenvalue with infinite multiplicity) between them. We describe all magnetic and electric fields for which the unperturbed flat band remains the flat band and when one splits into the small band of the continuous spectrum. Also we determine spectral asymptotics for small fields and solve inverse spectral problem.