$k$ on, $k<n$

Let $p=n-k$.

$$\left(\bar{x}_l-\alpha\right)\left( 1+p\bar{x}_{l}^c + k\bar{x}_{h}^c \right) = \sigma \bar{x}_{l}^c$$

$$p \bar{x}_{l}^{c+1} - \left( p\alpha +\sigma \right) \bar{x}_{l}^c + \left( 1+k \bar{x}_{h}^c \right)\bar{x}_{l} -$$

$$\alpha \left( 1+k \bar{x}_{h}^c \right) =$$ 0

$$k \bar{x}_{h}^{c+1} - \left( k\alpha +\sigma \right) \bar{x}_{h}^c + \left( 1+p \bar{x}_{l}^c \right)\bar{x}_{h} -$$

$$\alpha \left( 1+p \bar{x}_{l}^c \right) =$$ 0

Choosing for example the graded lexicographic order over $\mathbb{C}[{x}_{l},{x}_{h}]$, theorem 5.3.6 from Cox et al. (1996) shows that the system has a finite number of solutions, when $c$ is an integer.

We have

$$J_{i,i}=-1 + c\sigma x_{i}^{c-1}\frac{D-x_i^c} {D^2}$$

$$J_{i,j}=-c\sigma x_j^{c-1}\frac{x_i^c}{D^2}$$

If $x_i=x_j$,

$$J_{i,i}-J_{i,j}=-1 + c x_i^{c-1} \frac{\sigma}{D}=-1 + c\frac{x_i - \alpha}{x_i}$$

Consider the reordered Jacobian matrix, with $k$ variables "on" with a value $\bar{x_h}$, and $p$ "off" with a value $\bar{x_l}$ ($k+p=n$).

It follows from the analysis in section C.3 that the equilibrium can be stable only if $J_{i,i}-J_{i,j}<0$ (ie $x_i<\alpha \frac{c}{c-1}$), if the number of variables having value $x_i$ is strictly greater than 1.

Thus there are only two possible kinds of stable equilibria: all variables equal, in which case the equilibrium value is lower than $\alpha \frac{c} {c-1}$, or one higher than all the other ones (in which case the lower ones are lower than, and the higher one greather than $\alpha \frac{c} {c-1}$).