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| Lie's method of converting a second order scalar ordinary differential equation (ODE) to linear form by transformations of the independent and dependent variables (point transformations) had been extended to the third order. Separately, it had been shown that whereas for the second order ODEs there is a unique class of ODEs (with eight infinitesimal symmetry generators) for the third order there are three classes with four five and seven generators. Lie's linearization had also been extended to systems of second order ODEs and it had been found that for the two-dimensional system there are five classes (with five, six, seven, eight and fifteen generators). Another development used the analyticity of complex scalar ODEs to linearize two-dimensional systems. Based on the use of geometric methods, which not only convert the second order system to linear form but also directly provide the solutions for them, it produced three of the five classes (with six, seven and fifteen generators). Here we use the complex methods for the classification of two-dimensional systems of third order ODEs. It provides the classification of the subset of linearizable two-dimensional systems of third order ODEs that corresponds to a scalar complex third order ODE. Some examples are also given. |
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