I have an integral over a subspace of $\mathbb{R}^n \times \mathbb{R}^n$ with an integrand of the form $$\exp\left(-\frac{1}{2}\left[||u^2|| + \langle u, v \rangle + ||v||^2\right]\right)$$ The subspace is exactly the space for which $u_{i} = u_{n-i}$ (assume $n$ is even). In other words, $v$ is a true $n$-dimensional vector, whereas $u$ is two copies of an $n/2$ dimensional vector.

In order to evaluate this integral, I have been thinking about it as a density over ALL of $\mathbb{R}^n \times \mathbb{R}^n$ of two correlated Gaussians with covariance given by the block matrix $$\begin{bmatrix} A & B \\ B^t & C \end{bmatrix} $$ where $A,C$ are $n \times n$ identity matrices and $B$ is the block matrix $$\begin{bmatrix} I_{n/2} & I_{n/2} \\ 0 & 0 \end{bmatrix} $$ However, given that this is degenerate, I am having trouble finishing the computation. Would greatly appreciate any tips!