Prolongations of Vector Fields and the Invariants-by-Derivation Property

For any given vector field $X$ defined on some open set $M\subset \mathbb R^2$, we characterize the prolongations $X^*_n$ of $X$ to the nth jet space $M^{(n)}$, $n\geq 1$, such that a complete system of invariants for $X^*_n$ can be obtained by derivation of lower-order invariants. This leads to characterizations of $C^{\infty }$-symmetries and to new procedures for reducing the order of an ordinary differential equation.