  It is known for scalar ordinary differential equations, and for systems of
ordinary differential equations of order not higher than the third, that their
Lie point symmetry algebras is of maximal dimension if and only if they can be
reduced by a point transformation to the trivial equation $\mathbf{y}^{(n)}$=0.
For arbitrary systems of ordinary differential equations of order $n \geq 3$
reducible by point transformations to the trivial equation, we determine the
complete structure of their Lie point symmetry algebras as well as that for
their variational, and their divergence symmetry algebras. As a corollary, we
obtain the maximal dimension of the Lie point symmetry algebra for any system
of linear or nonlinear ordinary differential equations.
