On the non-integrability of the electro-dissolution of copper

In this work, we investigate integrability of three-dimensional systems for the copper electro-dissolution model, which are expressed as nonlinear ordinary differential equations$\dot{X}=Y,\dot{Y}=Z,\dot{Z}=-Z-\mu X-1.3Y+X^2-1.425Y^2+0.2XZ-0.01X^{2}Z.$ Where $X,Y$ and Z represent chemical concentrations, and μ is a real parameter. More precisely, we prove: first that the system has no polynomial, rational and Darboux first integrals and second that the system has no analytic first integrals in a neighborhood at the origin when $\mu \in [0,\frac{13}{10}]$ or $\mu \ne \frac{(13n^2+16n+13)n}{10{(n+1)}^3}$ where $n\in Z\setminus \left\{0\right\}$ as well as in a neighborhood at the equilibrium point $(\mu ,0,0)$ when $\mu =\frac{370}{13}\pm \frac{200\sqrt{3}}{13}$ or $\mu <0$ and $\frac{13{(\mu -10)}^2}{1000}+\mu >0$.