The bi-variate Cauchy distribution is given by $$ \mathbb{R}^2\ni(x,y)\mapsto\frac{1}{2\pi}\frac{1}{(x^2+y^2+1)^{1.5}}\,. $$
What is the name of the following probability distribution: $$ \mathbb{R}^2\ni(x,y)\mapsto\frac{1}{\pi}\frac{1}{(x^2+y^2+1)^{2}}\,? $$
Looks like a multivariate $t$-distribution with $\nu = 2$ degrees of freedom, mean $\mu = 0$ and shape matrix $\Sigma = 2I$:
https://en.wikipedia.org/wiki/Multivariate_t-distribution