How to integrate a specific set of variables from a Gaussian integrand?

Question

Let $x^T=\left(x_1,x_2,\ldots,x_N\right)$. I have the Gaussian function $\exp\left(-\frac{1}{2}x^TAx\right)$ where $A$ is positive definite and symmetric. I want to integrate this function with respect to $y$, where $y^T=\left(x_1,x_{1+i},x_{1+2i},\ldots,x_{1+ki}\right)$ with $1+ki<N$ and k an integer. An example would be tracing the odd numbered coordinates when $N=12$, $i=2$ and $k=5$, then $y^T=\left(x_1,x_3,x_5,x_7,x_9,x_{11}\right)$. Another example is when $N=12$, $i=3$ and $k=3$, then $y^T=\left(x_1,x_4,x_7,x_{10}\right)$. Then I want to evaluate the following integral (with every quantity real) \begin{align} \int_{-\infty}^{+\infty} \exp\left(-\frac{1}{2}x^TAx\right)dy \end{align} The motivation to solve this integral comes from working with Gaussian wave functions of a Hamiltonian describing a circular lattice, where I need to trace out some position variables $x_i$ according to this pattern rather than linearly, i.e,$\left(x_1,x_2,x_3\ldots, x_n\right)$ with $n<N$.