Let $k$ be any field (maybe need to be algebraically closed), and $n$ be a positive integer, $k[x_1,\ldots,x_n]$ be the polynomial ring. Then consider the equation $$\sum_{i=1}^n l_i(x)x_i^2=q(x)(x_1+\ldots+ x_n)$$ where $l_i(x)\in k[x_1,\ldots,x_n]$ is a homogeneous polynomial of degree $1$, and $q(x)\in k[x_1,\ldots,x_n]$ is a homogeneous polynomial of degree $2$. We want to find the solution $(l_i,q)$ for the equation above. Easy to see $$l_i=x_1+\ldots+ x_n$$ $$q=x_1^2+\ldots+ x_n^2$$ is a trivial solution (and any $\mathbb C$ multiple, i.e. $c_i l_i$ for $c_i\in k$). My question is
is there any other non-trivial solution?
In general I am concerned about the solution to $$\sum_{i=1}^n l_i(x)x_i^2=q(x)(\text{a general linear equation})$$ but let's see the special case first. I have no idea how to do this.
Edit
Robert's answer shows there are other solutions for $n<5$. Now I want to know the case $n=5$.
Algebraically closed is not an issue: this involves solving a linear system of equations for the coefficients of the $l_i$ and $q$, with integer coefficients.
You will find other nontrivial solutions. For example, for each $i$ you could have $l_i = x_1 + \ldots + x_n$, all other $l_j = 0$, $q(x) = x_i^2$.
In the case $n=3$ you could also take $l_1(x) = x_2 - x_3$, $l_2(x) = x_1$, $l_3(x) = -x_1$, $q(x) = x_1 x_2 - x_1 x_3$ (and similarly for permutations of the variables).
In the case $n=4$ you could take $$ \eqalign{l_{{1}}(x)&=x_{{2}}-x_{{4}}\cr l_{{2}}(x)&=x_{{1}}-x_{{3}}\cr l_{{3}}(x)&=-x_{{2}}+x_{{4}}\cr l_{{4}}(x)&=-x_{{1}}+x_{{3}}\cr q(x)&=x_{{1}}x_{{2}}-x_{{1}}x_{{4}}-x_{{2}}x_{{3}}+x_{{3}}x_{{4}}\cr}$$ (and again permutations of this).
I suspect this type of solution doesn't exist for $n\ge 5$.