  We show the total space of the canonical line bundle $\mathbb{L}$ of a
Kahler-Einstein manifold $X^n$ supports integrable $SU(n+1)$ structures, or
Calabi-Yau structures. The canonical real line bundle $L \subset \mathbb{L}$
over a minimal Lagrangian submanifold $M \subset X$ is calibrated in this
setting and hence can be considered as the special Lagrangian lift of $M$.
  For the integrable $G_2$ and $Spin(7)$ structures on spin bundles and bundles
of anti-self-dual 2-forms on self-dual Einstein 4-manifolds constructed by
Bryant and Salamon, minimal surfaces with vanishing complex quartic form
(super-minimal) admit lifts which are calibrated, i.e., associative,
coassociative or Cayley respectively. The lifts in this case can be considered
as the tangential lifts or normal lifts of the minimal surface adapted to the
quaternionic bundle structure.
