# Phase Portraits of Families VII and VIII of the Quadratic Systems
## 3 Proof of Statement (a) of Theorem 1  
### Finite Equilibrium Points  
We will determine the local phase portrait at the finite equilibrium points of the quadratic system (2).  
Assume that \(n\
eq 0\). If \(b^{2}-4 dn>0\), then the finite equilibrium points of system (2) are as follows:  
\[p_{\pm}=\left(0,\frac{-b\pm\sqrt{b^{2}-4 dn}}{2n}\right),\]  
The eigenvalues of the Jacobian matrix of system (2) at \(p_{\pm}\) are \(0\) and \(\pm\sqrt{b^{2}-4 dn}\). Thus, from Theorem 2.19 of [8], we have that \(p_{+}\) and \(p_{-}\) are semi-hyperbolic saddle-nodes.  
If \(b^{2}-4 dn<0\), there are no finite equilibrium points.  
If \(b^{2}-4 dn=0\), then \(d=b^{2}/(4n)\) and \(p_{+}=p_{-}=p=(0,-b/(2n))\). The Jacobian matrix of the differential system at \(p\) is  
\[\left(\begin{array}{cc}0&0\\ a-\frac{bm}{2n}&0\end{array}\right).\]  
If \(a-bm/(2n)\
eq 0\), then this equilibrium point is nilpotent, and from Theorem 3.5 of [8], this equilibrium point is a saddle-node.  
If \(a=bm/(2n)\), the linear part of the differential system at the equilibrium point \(p\) is identically zero, and the differential system becomes a homogeneous quadratic differential system. Using the results by [PERSON] in [14], who classified the phase portraits of all the homogeneous quadratic systems, we can see that the phase portraits of system (3) when \((m-1)^{2}-4\ell n>0\) are given in Figure 1, according to the sign of \(n\), If \((m-1)^{2}-4\ell n=0\), then the phase portraits of system (3) are given in Figure 2, determined by the sign of \(n\). Finally, if \((m-1)^{2}-4\ell n<0\), the phase portraits of system (3) are given in Figure 3, determined by the sign of \(n\).  
We assume that \(n=0\). In this case, if \(b\
eq 0\), there exists a unique equilibrium point, namely \(q=(0,-d/b)\), and the eigenvalues of the Jacobian matrix at \(q\) are \(0\) and \(b\). If \(b\
eq 0\), then \(q\) is a semi-hyperbolic saddle-node (by Theorem 2.19 of [8]). If \(b=0\) and \(d\
eq 0\), the differential system has no finite equilibria. If \(b=d=0\), then the system has a straight line filled with equilibria; we do not consider this kind of differential system because this case can be reduced to a linear differential system, involving the rescaling of the independent variable.  
In summary, we proved the following proposition.  
**Proposition 1**.: _Assume that \(n\
eq 0\)._  
1. _If_ \(b^{2}-4 dn>0\)_, the differential system (_2_) has two finite equilibria_ \(p_{\pm}\) _that are semi-hyperbolic saddle-nodes._
2. _If_ \(b^{2}-4 dn<0\)_, the differential system (_2_) has no finite equilibria._
3. \(b^{2}-4 dn=0\)_._ 1. _If_ \(a-bm/(2n)\
eq 0\)_, the differential system (_2_) has one finite equilibrium point_ \(p\) _that is a nilpotent saddle-node._ 2. \(a-bm/(2n)=0\)_._ 2. _If_ \((m-1)^{2}-4\ell n>0\)_, the phase portrait of the differential system (_2_) is topologically equivalent to the ones in Figure_ 1_, determined by the sign of_ \(n\)_._ 3. _If_ \((m-1)^{2}-4\ell n=0\)_, the phase portrait of the differential system (_2_) is topologically equivalent to the ones in Figure_ 2 _determined by the sign of_ \(n\)_._ 4. _If_ \((m-1)^{2}-4\ell n<0\)_, the phase portrait of the differential system (_2_) is topologically equivalent to the ones in Figure_ 3_, determined by the sign of_ \(n\)_._ _Assume that_ \(n=0\)_._ 4. _If_ \(b\
eq 0\)_, the differential system (_2_) has one finite equilibria_ \(q\)_, which is a semi-hyperbolic saddle-node._ 5. _If_ \(b=0\)_, then the differential system (_2_) has no finite equilibria if_ \(d\
eq 0\)_, and one straight line is filled with equilibria if_ \(d=0\)_._