  We present explicit constructions of complete Ricci-flat Kahler metrics that
are asymptotic to cones over non-regular Sasaki-Einstein manifolds. The metrics
are constructed from a complete Kahler-Einstein manifold (V,g_V) of positive
Ricci curvature and admit a Hamiltonian two-form of order two. We obtain
Ricci-flat Kahler metrics on the total spaces of (i) holomorphic C^2/Z_p
orbifold fibrations over V, (ii) holomorphic orbifold fibrations over weighted
projective spaces WCP^1, with generic fibres being the canonical complex cone
over V, and (iii) the canonical orbifold line bundle over a family of Fano
orbifolds. As special cases, we also obtain smooth complete Ricci-flat Kahler
metrics on the total spaces of (a) rank two holomorphic vector bundles over V,
and (b) the canonical line bundle over a family of geometrically ruled Fano
manifolds with base V. When V=CP^1 our results give Ricci-flat Kahler orbifold
metrics on various toric partial resolutions of the cone over the
Sasaki-Einstein manifolds Y^{p,q}.
