On certain types of point symmetries of systems of second-order ordinary differential equations

• The symmetry condition for systems of second order ODEs is refined for non-maximally symmetric systems.
• Existence and structure condition for certain types of symmetries are developed.
• Structure of subalgebras generated by symmetries are studied.
• Number of independent symmetries is obtained.

Existence criteria for some generic types of point symmetries of systems of n-second order ordinary differential equations are studied, specially in connection with the generation of semisimple subalgebras of symmetries belonging to the simple linear and orthogonal types, as well as their maximal dimension and rank. The structure of certain time-dependent symmetries, in particular scaling symmetries, are also studied, and the structure of the subalgebras they span determined. Generic examples illustrating the procedure are given.