  Noether's theorem relates constants of motion to the symmetries of the
system. Here we investigate a manifestation of Noether's theorem in
non-Hermitian systems, where the inner product is defined differently from
quantum mechanics. In this framework, a generalized symmetry which we term
pseudo-chirality emerges naturally as the counterpart of symmetries defined by
a commutation relation in quantum mechanics. Using this observation, we reveal
previously unidentified constants of motion in non-Hermitian systems with
parity-time and chiral symmetries. We further elaborate the disparate
implications of pseudo-chirality induced constant of motion: It signals the
pair excitation of a generalized "particle" and the corresponding "hole" but
vanishes universally when the pseudo-chiral operator is anti-symmetric. This
disparity, when manifested in a non-Hermitian topological lattice with the
Landau gauge, depends on whether the lattice size is even or odd. We further
discuss previously unidentified symmetries of this non-Hermitian topological
system, and we reveal how its constant of motion due to pseudo-chirality can be
used as an indicator of whether a pure chiral edge state is excited.
