Line bundles on spectral curves and the generalised Legendre transform construction of hyperkähler metrics

An analogue of the correspondence between GL(k)$GL\left(k\right)$-conjugacy classes of matricial polynomials and line bundles is given for K$K$-conjugacy classes, where K⊂GL(k)$K\subset GL\left(k\right)$ is one of the following: maximal parabolic, maximal torus, GL(k−1)$GL(k-1)$ embedded diagonally. The generalised Legendre transform construction of hyperkähler metrics is studied further, showing that many known hyperkähler metrics (including the ones on coadjoint orbits) arise in this way, and giving a large class of new (pseudo-)hyperkähler metrics, analogous to monopole metrics.