  Monopoles are solutions of an SU(2) gauge theory in $\mathbb{R}^{3}$
satisfying a lower bound for energy and certain asymptotic conditions, which
translate as topological properties encoded in their charge. Using methods from
integrable systems, monopoles can be described in algebraic-geometric terms via
their {spectral curve}, i.e. an algebraic curve, given as a polynomial P in two
complex variables, satisfying certain constraints. In this thesis we focus on
the Ercolani-Sinha formulation, where the coefficients of P have to satisfy the
Ercolani-Sinha constraints, given as relations amongst periods. A particular
class of such monopoles is studied, namely charge 3 monopoles with a symmetry
by $C_{3}$, the cyclic group of order 3. This class of cyclic 3-monopoles is
described by the genus 4 spectral curve $\hat{X}$, subject to the
Ercolani-Sinha constraints: the aim of the present work is to establish the
existence of such monopoles, which translates into solving the Ercolani-Sinha
constraints for $\hat{X}$.
