If it exists,a surface S containing the affine plane as a proper open subset U and equipped with a morphism to the affine plane,whose restriction to U is étale,provides a counter-example to the Jacobian Conjecture.The class of surfaces containing the affine plane is very large while the class of available methods to exclude the existence of morphisms of the previous type is very limited.In this talk,I will mainly focus on a special sub-class of surfaces arising as total spaces of locally trivial affine bundles over the affine line with a double origin.This sub-class contains for instance the smooth affine quadric and a family of surfaces on which the problem has been already considered during the last decades successively by Peretz and Wright,using different methods.Quite deceptively,it turns out that all known positive results for this sub-class can be re-interpreted in terms of the non-vanishing of a certain class in the algebraic De Rham cohomology of the surface.After reviewing this algebrodifferential argument,Ill try to propose some potential alternative strategies to handle the cases in which this naive obstruction class vanishes.Here the problem becomes more subtle,in particular,I will give an example of surface U with the same algebro-differential invariants as the affine plane,a surface S in the class considered containing U as a proper open subset and equipped with a proper ′etale morphism S→ U(to my knowledge,this surface U equipped with the induced ′etale endomorphism also provides a new counter-example to the Generalized Jacobian Conjecture in the sense of Miyanishi).