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We theoretically study exotic superconducting phases in graphenelike single-sheet material BC$_3$ doped to its type-II Van Hove singularity whose saddle point momenta are {\it not} time-reversal-invariant. From combined renormalization group analysis and RPA calculations, we show that the dominant superconducting instability induced by weak repulsive interactions is in the time-reversal-invariant …