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We establish the existence of positive normalized (in the $L^2$ sense)solutions to non-variational weakly coupled elliptic systems of $\ell$equations. We consider couplings of both cooperative and competitive type. Weshow the problem can be formulated as an operator equation on the product of$\ell$ $L^2$-spheres and apply a degree-theoretical argument on this product toobtain existence.

Normalized solutions to a non-variational Schr\"odinger system