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This work analyses types of group actions on families of $t$-dependent vectorfields of a particular class, the hereby called quasi-Lie families. We devisemethods to obtain the defined here quasi-Lie invariants, namely a kind offunctions constant along the orbits of the above-mentioned actions. Ourtechniques lead to a deep geometrical understanding of quasi-Lie schemes andquasi-Lie systems giving rise to several new results. Our achievements areillustrated by studying Abel and Riccati equations. We retrieve the Liouvilleinvariant and study other new quasi-Lie invariants of Abel equations. SeveralAbel equations with a superposition rule are described and we characterise Abelequations via quasi-Lie schemes.

Quasi-Lie families, schemes, invariants and their applications to Abel equations