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We investigate in underdoped cuprates possible coexistence of the superconducting order at zero momentum and pair density wave (PDW) at momentum Q =(π,π) in the presence of a Néel order.By symmetry,the d-wave uniform singlet pairing dS0 can coexist with the d-wave triplet PDW dTQ,and the p-wave singlet PDW pSQ can coexist with the p-wave uniform triplet pT0.At half filling,we find that the novel pSQ + pT0 state is energetically more favorable than the dS0 + dTQ state.At finite doping,however,the dS0 + dTQ state is more favorable.In both types of states,the variational triplet parameters dTQ and pT0 are of secondary significance.Our results point to a fully symmetric Z2 quantum spin liquid with spinon Fermi surface in proximity to the Néel order at zero doping,which may not be adiabatically connected to the d-wave singlet superconductivity at finite doping with intertwining d-wave triplet PDW fluctuations and spin moment fluctuations.The results are obtained by variational quantum Monte Carlo simulations.