I would like to add a further analogy and an application for the case of the complex complete flag manifold $G/B$. In this case $ X = G = SU(N)$ is isomorphic (as a homogeneous space) to its principal homogeneous space: the Stiefel manifold $V_{n-1}(\mathbb{R}^n)$, as given in the following wikipedia page.

In both cases of the sphere and the Stiefel manifold, they can be given as (intersection of) quadrics. in $\mathbb{C}^n$ (the stiefel manifold can be identified with the space of full rank $n\times (n-1)$ matrices satisfying $\bar{M}^t M = 1$). Of course, the case of the standard Hopf fibration (X = S^3, M = S^2) is mutual to both cases.

This construction can be applied to perform integrations over the $SU(n)$ invariant measures
of the complex projective spaces and complete flag manifolds: Integration over spheres and Stiefel manifolds is relatively straightforward because they can be performed on quadric constraint surfaces in $\mathbb{C}^n$. The integrals can be converted to (a series of) Gaussian integrals by means of Fourier transforms.

Functions on the complex projective spaces and complete flag manifolds can be extended to functions on the total manifolds, the sphere and the Stiefel manifold by defining them to be constants along the torus fibers.

In the case of the complex projective spaces, the sphere is identified with the space of
unit vectors in $\mathbb{C}^n$ and the complex projective space with the projectors onto these vectors. Thus any function on the sphere which depends solely on the projectors of the unit vector is an extension of a function on the complex projective space and its integral on the sphere is proportional to the integral of the original function on the complex projective space.

In the case of the complete flag manifold, the Stiefel manifold may be considered as the space of orthonormal frames in $C^n$, (the column vectors of $M$) and the flag manifold as the space of projectors onto these vectors. Again, functions on the Stiefel manifold depending solely on the projectors of the frame vectors are extensions of functions on the flag manifolds.