Affine surfaces with a huge group of automorphisms

We describe a family of rational affine surfaces S with huge groups of automorphisms in the following sense: the normal
subgroup Aut(S)alg of Aut(S) generated by all algebraic subgroups of Aut(S) is not generated by any countable family
of such subgroups, and the quotient Aut(S)/Aut(S)alg cointains a free group over an uncountable set of generators.