Construct bivariate symmetric polynomials $f(x,y) = f(y,x) \ge 0$ over $[0,1]^2$, with $f(1,y) = f(x,1)=0$, such that the univariate marginal distributions are both proportional to
$$(1-u^2)^4$$, where $u$ is either $x$ or $y$
and the "diagonal" $f(x,x) = f(y,y)$ is proportional to
$$(1-u)^5$$
Also, the same form of problem with marginals
$$(1-u^2)^6$$
and diagonal
$$(1-u)^9 (1+8 u)$$
And a third case requiring the marginals
$$(1-u^2)^8$$
and diagonal
$$(1-u)^{11} (1+u (11+40 u))$$
These problems have a quantum-information-theoretic relevance pertaining to separability probability calculations.