  In this paper, we study residual symmetries in the lepton sector. Our first
concern is the symmetry of the charged lepton mass matrix in the basis where
the Majorana neutrino mass matrix is diagonal, which is strongly constrained by
the requirement that the symmetry group generated by residual symmetries is
finite. In a recent work R. M. Fonseca and W. Grimus found that there exists a
set of constraint equations that can be completely solved, which is essential
in their approach to the classification of lepton mixing matrices that are
fully determined by residual symmetries. In this paper, a method to handle
trigonometric Diophantine equations is introduced. We will show that the
constraint equations found by Fonseca and Grimus can also be solved by this
method. Detailed derivation and discussion will be presented in a
self-contained way. In addition, we will also show that, in the case where
residual symmetries satisfy a reality condition, this method can be used to
solve the equation constraining parameters in the symmetry assignment that
controls the group structure generated by residual symmetries and is directly
related to mixing matrix elements.
