It is well - known that if $x=(x_1,...,x_n)^T\sim{N(0, \sigma^2I)}$, then its normalized version is uniformly distributed on the unit $n-1$ - sphere:
$$ y:=\frac{x}{||x||_2}\sim{\text{Uniform}}(S_{n-1}(1)). $$
From this it is easy to derive both the conditional and marginal distributions of $y$ as well.
Now, the case that I'm interested in is the non-spherical case: $x\sim{N(0, D)}$, where $D$ can assumed to be diagonal. In this case it is again easy to derive by integration that $$ p_{\frac{x}{||x||_2}}(t)=|D|^{-1/2}(t^TD^{-1}t)^{-n/2} $$ with respect to the uniform measure on a unit $n-1$-sphere. However, I struggle with deriving marginal ($y_1$) and/or conditional ($(y_2,...,y_n)|y_1$) distributions of this vector since I quickly run into complicated integrals in spherical coordinates. I would highly appreciate any leads and/or even any available bounds/series expansions/approximations of these distributions.
Thank you!