Generalized gaussian integral on a sphere

Question

I am interested in the calculation of the following integral: \begin{equation} I=\int_{\sum_{i=1}^Nx_i^2=R^2} \mathrm d x \;e^{-\frac 12 x^T A x} \end{equation} where $A\in \mathbb M_N(\mathbb R)$ is a real symmetric matrix. I tried to rewrite the condition as a Dirac delta \begin{equation} \delta\left(\sum_{i=1}^Nx_i^2-R^2\right) = \frac 1 {2\pi} \int_{\mathbb R}\mathrm d k\; e^{ik(\sum_{i=1}^Nx_i^2-R^2)} \end{equation} but this adds a purely complex diagonal term to the matrix $A$, which is then not Hermitian anymore. Can this integral be solved?