algebra precalculus - Functional equation with $f(1)$ integer

Here is a nice problem: Let $f:R\rightarrow R$ be a function, R is the set of real numbers, satisfying the following properties: $ f(1)$ is an integer and
$xf(y)+yf(x)=(f(x+y))^2-f(x^2)-f(y^2)$, for all x, y real numbers.
$f(x)=0$ is a solution, another is $ f(x)=x $. These are all solutions?

better asking: determine all functions that satisfy the 2 conditions. I would like to see a complete solution! Thank you!