  We show that the monodromy of a spherical conical metric is reducible if and
only if it has a real-valued eigenfunction with eigenvalue 2 in the holomorphic
extension of the associated Laplace--Beltrami operator. Such an eigenfunction
produces a meromorphic vector field, which is then related to the developing
maps of the conical metric. We also give a lower bound of the first nonzero
eigenvalue, and a complete classification of the eigenspace dimension depending
on the monodromy. This paper can be seen as a new connection between the
complex analysis method and the PDE approach in the study of spherical conical
metrics.
