On solvable Lie algebras and integration method of ordinary differential equations.

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.