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A large class of solutions of the Einstein-conformal scalar equations in D=2+1 and D=3+1 is identified. They describe the collisions of asymptotic conformal scalar waves and are generated from Einstein-minimally coupled scalar spacetimes via a (generalized) Bekenstein transformation. Particular emphasis is given to the study of the global properties and the singularity structure of the obtained solutions. It is shown, that in the case of the absence of pure gravitational radiation in the initial data, the formation of the final singularity is not only generic, but is even inevitable.

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