We study the equivariant cohomology of moduli spaces of objects in the derived category of elliptic surfaces in order to find new examples of infinite dimensional quantum integrable systems and their geometric representation theoretic interpretation in enumerative geometry. This problem is related to a program to understand the cohomological and K-theoretic Hall algebras of holomorphic symplectic surfaces and to understand how it related to the Donaldson-Thomas theory of threefolds fibered in those surfaces. We use the theory of noncommutative deformations of Poisson surfaces and especially van den Berg’s noncommutative P1 bundles as well as Rains’s analysis of moduli theory for quasi-ruled noncommutative surfaces as well as the theory of Bridgeland stability conditions and their relative versions to understand equivariant deformations and birational transformations of Hilbert schemes of points on equivariant elliptic surfaces. The moduli spaces are closely related to elliptic versions of classical integrable systems. We also use these moduli spaces to construct vertex algebra representations of extensions of toroidal extended affine algebras on their equivariant cohomology, building on work of Eswara-Rao–Moody–Yokonuma, of Billig, and of Chen–Li–Tan on vertex representations of toroidal algebras, full toroidal algebras, and toroidal extended affine algebras. Using Fourier-Mukai transforms and their relative action on families of dg-categories we study the relationship between automorphisms of toroidal extended affine algebras and families of derived equivalences on dg categories, in particular finding a relativistic (difference) generalization of the Laumon-Rothstein deformation of the Fourier-Mukai duality. Finally, using the above analysis we extend the construction of Maulik–Okounkov’s stable envelopes to moduli of framed torsionfree sheaves on noncommutative surfaces in some cases and use this to study coproducts on associated algebras assigned to elliptic surfaces with applications to understanding new representation theoretic structures in the Donaldson-Thomas theory of local curves.