# Some Dynamic Aspects of a Sextic Galactic Potential in a Rotating Reference Frame
## 2 Integrability Analysis
The potential \(V_{0}(x,y)\) is  
\[\dot{x} =p_{x}+\frac{\Omega}{\alpha^{2}}y, \tag{6a}\]1  
The solutions to the algebraic equation \(\
abla V_{0}(c_{1},c_{2})=(c_{1},c_{2})\) are as follows:  
(1) \(c_{1}=0,c_{2}=\frac{1}{\sqrt[4]{b}}\), where \(b>0\);  
(2) \(c_{1}=\frac{1}{\sqrt[4]{a}},c_{2}=0\), where \(a>0\). For the first solution, the non-integrability coefficient is \(\sigma=\frac{d}{b}\), while the non-integrability coefficient corresponding to the second solution is \(\sigma=\frac{c}{a}\). Taking into account [PERSON]'s theorem (Theorem 2), the Hamiltonian in Equation (8) is non-integrable if at least one of the following conditions is satisfied:  
\[ \begin{split}\frac{d}{b}\in]-\infty,0[\cup]j(3j-2),j(3j+2)[,& j\in\mathbb{N},\\ \frac{c}{a}\in]-\infty,0[\cup]j(3j-2),j(3j+2)[,& j \in\mathbb{N}.\end{split}  \tag{12}\]  
The next Theorem follows from Theorem 1 and Theorem 2:  
**Theorem 3**. : _The galactic Hamiltonian in Equation (1) with the sextic galactic potential in Equation (2) is non-integrable if at least one of the conditions in Equation (12) is satisfied._  
**Remark 1**. : _Theorem 3 provides the sufficient conditions for non-integrability. This means that if one of the conditions in Equation (2) is fulfilled, then the Hamiltonian in Equation (1) is non-integrable, and no further investigation is required. On the other hand, if neither conditions in Equation (2) hold, then the Hamiltonian in Equation (1) may be either integrable or non-integrable. In such a case, verifying integrability requires finding an additional integral functionally independent of the Jacobi integral in Equation (4)._  
It is worth mentioning that [PERSON] introduced a theorem [34] based on studying the solvability of the differential Galois group associated with the normal variational equation, which can be applied effectively to give the necessary conditions for integrability and sufficient conditions for the non-integrability for a wide class of non-homogeneous potentials, including the galactic potential in Equation (2) as a special case. Special cases:  
(1) If \(a=b=c=d\), then the Hamiltonian in Equation (1) takes the form below in polar coordinates \((r,\theta)\):  
\[H=\frac{1}{2}(p_{r}^{2}+\frac{1}{r^{2}}p_{\theta}^{2})-\Omega p_{\theta}+\frac{ a}{6}r^{6}+\frac{r^{2}}{2}. \tag{13}\]The Hamiltonian in Equation (13) is integrable since it has two independent first integrals of the motion. The first is the momentum \(p_{\theta}=\alpha\) corresponding to the cyclic coordinate \(\theta\), with the second being the Jacobi integral, which takes the form  
\[\frac{1}{2}p_{r}^{2}+\frac{\alpha^{2}}{2r^{2}}+\frac{a}{6}r^{6}+\frac{r^{2}}{2} =E. \tag{14}\]  
It should be noted that the constants \(a,b,c\), and \(d\) in the present case do not satisfy Theorem 3 since \(\frac{d}{b}=\frac{c}{a}=1\) does not fall in the non-integrability regions. 1. If \(c=d=0\) and \(\Omega=0\), then the Hamiltonian in Equation (1) is separable into Cartesian coordinates. As a result of the separability of the Hamiltonian, the additional integral is quadratic in momentum. The integrals of the motion for this case are \[ \begin{split}\frac{1}{2}p_{x}^{2}+\frac{1}{2}x^{2}+\frac{a}{6}x^ {6}&=\gamma,\\ \frac{1}{2}p_{y}^{2}+\frac{1}{2}y^{2}+\frac{b}{6}y^{6}& =E-\gamma,\end{split} \] (15) where \(\gamma\) is a constant. It should be noticed that the non-integrability coefficient \(\sigma\) is zero, and it does not fall within the non-integrability regions. We present the Poincare surface of a section for the Hamiltonian in Equation (1) with the potential function in Equation (2) when the parameters \(a,b,c\), \(d\), and \(\Omega\) take two different values. One of them satisfies Theorem 3, and the other does not agree with it. When \(a=b=c=d\), and \(\Omega\) is arbitrary, the non-integrability coefficient \(\sigma=1\) is outside the non-integrability region. The Poincare surface of the section, which is illustrated by Figure 1a, seems to be regular and gives some indications about the integrability of the problem.