  Some more general "inheritance conditions" have been found for a given set of
symmetry generators $\{\mathbf{Z}_{\bar{l}}\}$ acting on some set of coupled
ordinary differential equations, once the "first integration method" has been
applied upon some Abelian sub-algebra of generators $\{\mathbf{Z}_{i}\}$ for
the system of equations to be reduced. We have proven the following theorem:
all the generators of some solvable sub-algebra can be used via the "first
integration method" in order to reduce (the order) and/or (possibly) integrate
the system of equations. The specific order in which the step by step reduction
process has to be performed, and the condition to obtain the analytic solution
solely in terms of quadratures, are also provided. We define the notion
"optimum" for some given maximal, solvable sub-algebra and prove a theorem
stating that: this algebra leads to the most profitable way of using the
symmetry generators for the integration procedure at hand.
