  A spherical conical metric $g$ on a surface $\Sigma$ is a metric of constant
curvature $1$ with finitely many isolated conical singularities. The
uniformization problem for such metrics remains largely open when at least one
of the cone angles exceeds $2\pi$. The eigenfunctions of the Friedrichs
Laplacian $\Delta_g$ with eigenvalue $\lambda=2$ play a special role in this
problem, as they represent local obstructions to deformations of the metric $g$
in the class of spherical conical metrics. In the present paper we apply the
theory of multivalued harmonic maps to spheres to the question of existence of
such eigenfunctions. In the first part we establish a new criterion for the
existence of $2$-eigenfunctions, given in terms of a certain meromorphic data
on $\Sigma$. As an application we give a description of all $2$-eigenfunctions
for metrics on the sphere with at most three conical singularities. The second
part is an algebraic construction of metrics with large number of
$2$-eigenfunctions via the deformation of multivalued harmonic maps. We provide
new explicit examples of metrics with many $2$-eigenfunctions via both
approaches, and describe the general algorithm to find metrics with arbitrarily
large number of $2$-eigenfunctions.
