We consider a class, denoted by Q, of the nonlinear control systems which can be densely represented as a subsystem of a certain kind of quadratic system, namely a quadratic target. We say that a system in Q undergoes a globally exact quadratization. Here "globally" adds up to a slight extension of the notion of C-infinity immersion (of systems), namely a dense immersion, which amounts to saying that it is defined on the whole manifold of the system states, except possibly a zero-measure set. It is proven that the class Q includes all systems characterized by vector fields whose components are analytic integral closed-form functions (ICFFs). The result is first proven for algebraic system functions, by means of a constructive proof, and next extended up to analytic ICFFs. For nonanalytic ICFFs a weaker result is proven as well. Also the case of a partially observed system is considered, as well as the internal structure of every quadratic representation, which is proven to be always a feedback interconnection of bilinear systems. Finally, examples are presented for which the constructive proof given earlier is turned into a quadratization algorithm, which can be carried out by hand, and the resulting differential equations of the quadratic representation are presented.