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Let $\phi(x,y)=ax^2+2bxy+cy^2$ be a binary quadratic form with integer coefficients.

Pall proved in 1951 that $\phi(x,y)$ can be written as the sum of two squares of two linear forms if and only if $ac-b^2=m^2$ is a perfect square and $r_2(d)>0$, where $r_2(d)$ denotes the number of representations of $d=\gcd(a,2b,c)$ as the sum of two squares.

Moreover, the number of representations of $\phi(x,y)$ as a sum of the squares of two linear forms is exactly $2r_2(d)$ if $ac-b^2=m^2>0$. Williams gave in 1972 a simpler proof of this result.

My question is whether a similar result is known for ternary quadratic forms. More precisely, let

$\psi(x,y,z)=ax^2+by^2+cz^2+2dxy+2exz+2fyz$ be such a form with integer coefficients.

Are there any necessary and sufficient conditions on the coefficients which guarantee that $\psi(x,y,z)$ can be written as

$\psi(x,y,z) =(a_1x+b_1y+c_1z)^2+(a_2x+b_2y+c_2z)^2$ for some integers $a_i, b_i, c_i$?

Moreover, assuming that $\gcd(a,b,c,d,e,f)=1$, is it possible that $\psi(x,y,z)$ can be written as a sum of two squares in more than one way?

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