On the symmetry solutions of two-dimensional systems not solvable by standard symmetry analysis

A class of two-dimensional systems of second-order ordinary differential equations is identified in which a system requires fewer Lie point symmetries than required to solve it. The procedure distinguishes among those which are linearizable, complex-linearizable and solvable systems. We also present the underlying concept diagrammatically that provides an analogue in ${\mathrm{\Re }}^3$ of the geometric linearizability criteria in ${\mathrm{\Re }}^2$.