Let $C$ be a smooth quasi-projective connected curve over the complex numbers.

Can one classify all reductive group schemes over $C$?

Certainly, you have the trivial ones (coming from pulling-back those living over the complex numbers).

I can give a cohomological interpretation of this set as in Is every reductive group scheme etale locally trivial?, but I'm interested in a concrete description.

I would already be very happy to see some "non-trivial" examples of reductive group schemes over the affine line.