Algebraic recasting of nonlinear systems of ODEs into universal formats

It is sometimes desirable to produce for a nonlinear system of ODEs a new representation of simpler structural form, but it is well known that this goal may imply an increase in the dimension of the system. This is what happens if in this new representation the vector field has a lower degree of nonlinearity or a smaller number of nonlinear contributions. Until now both issues have been treated separately, rather unsystematically and, in some cases, at the expense of an excessive increase in the number of dimensions. We unify here the treatment of both issues in a common algebraic framework. This allows us to proceed algorithmically in terms of simple matrix operations.