A nonlinear formulation of differential Galois theory was discussed here and here for three dimensional nonlinear systems (proof is on pages 6 – 10). For a two dimensional system, the following system of differential equations was considered: \begin{equation} \begin{split} \dot{x}_1&=X_1(x_1,x_2),\\ \dot{x}_2&=X_2(x_1,x_2), \end{split} \end{equation} such that $X_1,X_2$ are polynomial functions and $X_1(0,0)\neq0$. Let $K$ denote the differential field of rational functions, with constant field $\mathbb{C}$. The system is Liouvillian integrable iff the differential Galois group of $dx_2/dx_1=X_2(x_1,x_2)/X_1(x_1,x_2)$ over $K$ at $(0,0)$ is solvable. For the four dimensional system \begin{equation} \begin{split} \dot{x}&=X,\\ \dot{X}&=\sigma(Y-X-kx),\\ \dot{Y}&=-Y+rX-XZ,\\ \dot{Z}&=-bZ+XY. \end{split} \end{equation} does an equivalent formulation exist to determine whether Liouvillian integrability (or solvability by quadratures) holds where $k,r,\sigma\in\mathbb{R}^+$ and $b=1$? Any help would be much appreciated.
Note that when $k=0$, we have the closed system. \begin{equation} \begin{split} \dot{X}&=\sigma(Y-X),\\ \dot{Y}&=-Y+rX-XZ,\\ \dot{Z}&=-bZ+XY. \end{split} \end{equation}
As a corollary of Equation 6 and 12 in the second linked paper, the Lorenz-like system is at least partially integrable when \begin{equation} 4\sigma r=b^2\frac{(2n+1)^2}{4}-(\sigma-1)^2, \end{equation} for all $n\in\mathbb{Z}^+$.
The following theorem of Ishii may help for a complete integrability criterion. Assume that the system has a balance $\{{a},{p}\in\mathbb{Q}^4\}$ for which the Kovalevskaya matrix \begin{equation} K=JV_{a}-\mathop{\mathrm{diag}}(p), \end{equation} is semi-simple (such that $JV_{a}$ denotes the Jacobian of the system of ODEs at $a$). If the system is completely integrable, then the local general series are Puiseux series.
