Multivariate integral of the Dirac delta distribution $\int_\mathbb{S}\int_\mathbb{S}f(x,y)\delta(x-y-a)dxdy$

I am confused about the multivariate integral of the Dirac delta distribution $\int_\mathbb{S}\int_\mathbb{S}f(x,y)\delta(x-y-a)dxdy$, where $\mathbb{S}\subset\mathbb{R}^2$ is a unit circle. For any $a\in\mathbb{R^2}$ in the ball $B(0,2)\subset\mathbb{R}^2$, should it be $\int_\mathbb{S}f(y+a,y)dy$ or $\int_{\mathbb{S}\cap\{y: y+a\in\mathbb{S}\}}f(y+a,y)dy$? Is the latter zero?There are two pairs of root of $x-y-a$, say $(x_1,y_1)$ and $(x_2,y_2)$. Should the original integral be $f(x_1,y_1)+f(x_2,y_2)$?