I am working with an equation to find the singular points in $\mathbb P^2 (\mathbb C)$ .
Basically after taking the partial derivatives and doing some manipulations it reduces to
$$y^2 + (2-k)xz +(k-2)yz -x^2 =0$$ $$z^2 +(2-k)xy +(k-2)xz -y^2=0$$ $$x^2+(2-k)yz+(k-2)xy-z^2=0$$
I can see that for $k=2 , 3$ the solution exists .
But I wonder if I can show that the solution doesn't exist for other $k\in \mathbb C$ ?
Thanks for your hints and helps .