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?

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  ?