# Comparing the variance of paired observations

I have $N$ paired observations ($X_i$, $Y_i$) drawn from a common unknown distribution, which has finite first and second moments, and is symmetric around the mean.

Let $\sigma_X$ the standard deviation of $X$ (unconditional on $Y$), and $\sigma_Y$ the same for Y. I would like to test the hypothesis

$H_0$: $\sigma_X = \sigma_Y$

$H_1$: $\sigma_X \neq \sigma_Y$

Does anyone know of such a test? I can assume in first analysis that the distribution is normal, although the general case is more interesting. I am looking for a closed-form solution. Bootstrap is always a last resort.