Suppose that $H(x,y,z)$ is a homogeneous polynomial of degree $n$, i.e., $H(x,y,z)=\sum_{i=1}^{k} a_i x^{\alpha_i} y^{\beta_i} z^{\gamma_i}$ s.t. $\alpha_i + \beta_i + \gamma_i = n, \forall i$, I need to find a bound on the number of isolated solutions (i.e., the ones that are the only one in an open neighborhood) of the following system of equations: $$\sum_{i=1}^{k} a_i \alpha_i x^{\alpha_i-1} y^{\beta_i} z^{\gamma_i} - n \sum_{i=1}^{k} a_i x^{\alpha_i+1} y^{\beta_i} z^{\gamma_i} = 0 ,$$
$$\sum_{i=1}^{k} a_i \beta_i x^{\alpha_i} y^{\beta_i-1} z^{\gamma_i} - n \sum_{i=1}^{k} a_i x^{\alpha_i} y^{\beta_i+1} z^{\gamma_i} = 0 ,$$
$$\sum_{i=1}^{k} a_i \gamma_i x^{\alpha_i-1} y^{\beta_i} z^{\gamma_i-1} - n \sum_{i=1}^{k} a_i x^{\alpha_i+1} y^{\beta_i} z^{\gamma_i+1} = 0 .$$
This is related to my previous question on the number of isolated rest points of a vector field. Any idea is greatly appreciated!