ThispaperusesPoincaréformalismtoextendtheLevi-Civitatheoremtocopewithnonholonomicsys-temsadmittingcertaininvariantrelationswhoseequationsofmotioninvolveconstraintmultipliers.Sufficientcondi-tionsallowingsuchextensionareobtainedand,asanapplicationofthetheoryageneralizationofRouth’smotionispresented.