I will introduce a class of infinite-dimensional (ind-)algebraic groups closely related to the group of polynomial automorphisms of the affine plane. Originating in the theory of integrable systems these groups have an interesting structure and share many properties with finite-dimensional semisimple affine algebraic groups. In particular, we prove an infinite-dimensional analogue of the classical theorem of R. Steinberg giving an abstract characterization of Borel subgroups in affine algebraic groups, and we will use this last result to actually classify Borel subgroups up to conjugation. The proofs are not entirely geometric nor algebraic: the crucial ingredient is the Friedland-Milnor analytic classification of polynomial automorphisms in C^2 (and its recent algebraic refinement due to S. Lamy). As a motivation I will explain a relation to the Jacobian Conjecture and its ‘quantum’ counterpart — the Dixmier Conjecture.