Do the relations between Galois groups and solutions to polynomial equations with one variable have a counterpart between Lie groups and solutions to differential equations?

Do the relations between algebraic variaties  and solutions to polynomial equations have a counterpart between differential manifolds and solutions to differential equations?

Is there a similar theorem/counterpart like Matiyasevich’s theorem for differential equations?

Any other parallel theorem, idea, theory exist for  differential equations as those for polynomial equations?