Nearly Kähler and special Kähler manifolds

Question

We know that the most important example of a nearly Kähler manifold is the sphere $S^{6}$ and that $(\nabla_{X}J)Y=-(\nabla_{Y}J)X$ is valid in this case (J - an almost complex structure). Similar condition is valid on special Kähler manifolds, $(\nabla_{X}J)Y=(\nabla_{Y}J)X$, but in this case $\nabla$ is not Levi-Civita. Is there a similar example of a special Kähler manifold, so that we can calculate on special Kähler manifolds using it?