  Binary compounds $A$Bi ($A$ = K, Rb, Cs) crystallize in P2$_1$/c structure
containing both clockwise and anticlockwise chiral chains of Bi atoms.
Electronic band structure exhibits the insulating nature of these systems, with
the band gap about $0.25$ eV. The presented study of dynamical properties
confirm a stability of the system with P2$_1$/c symmetry. The crystal structure
contains the quasi-one-dimensional Bi chains, exhibiting four-fold-like
rotational ``local'' symmetry. Nevertheless, the system formally posses
two-fold rotational symmetry. Independently of the absence of the three-fold
(or higher) rotational symmetry axes for the whole crystal, the chiral modes
propagate along the Bi atom chains in these systems. We discuss basic
properties of these modes in monoatomic chiral chains. We show that the
two-fold rotational symmetry axis affects the main properties of the chiral
phonons, which are not realized at the high-symmetry points, but along some
paths between them in the reciprocal space. In addition, in the doped system,
the chiral phonons possess non-zero total angular momentum.
