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The problem of exact (and global) bilinearization of nonlinear delay systems is considered for an algebraic system's function in two dimension. It is proven that a nonlinear map (bilinearizing map) can be defined on the functional space of state's trajectories such that the image-system, defined through this map, of the original system, is in general a bilinear hybrid system, evolving in the same state-space as the original, but switching among a family of bilinear delayed systems. For constant delays, equal for each state component, such hybrid system reduces to switching among just two bilinear systems working on two disjoint intervals (the second interval unbounded) partitioning the original system's time domain. Finally, an algorithm is defined which allows the recursive inversion of the bilinearizing map thus recovering the original system's state, and proving the equivalence of the bilinear representation at issue with the original nonlinear system. The paper is a first step towards a general exact bilinearization of nonlinear delayed systems having an any-order algebraic, and even analytical, system's functions.