We classify all possible charge-3 monopole spectral curves with non-trivial
automorphism group and within these identify those with elliptic quotients. By
focussing on elliptic quotients the transcendental constraints for a monopole
spectral curve become ones regarding periods of elliptic functions. We
construct the Nahm data and new monopole spectral curves with D6β and V4β
symmetry discovering a previously-unknown (to us) integrable complexification
of Euler's equations. Extensions of our approach to higher charge and
hyperbolic monopoles are discussed