  The class of 2-regular matroids is a natural generalisation of regular and
near-regular matroids. We prove an excluded-minor characterisation for the
class of 2-regular matroids. The class of 3-regular matroids coincides with the
class of matroids representable over the Hydra-5 partial field, and the
3-connected matroids in the class with a $U_{2,5}$- or $U_{3,5}$-minor are
precisely those with six inequivalent representations over GF(5). We also prove
that an excluded minor for this class has at most 15 elements.
