Several intermediate states of the reaction channels $e^{+}e^{-} rightarrow pi^{+} pi^{-} D^{0} bar{D}^{0}$ and $e^{+}e^{-} rightarrow pi^{+} pi^{-} D^{+}D^{-}$ are studied using the data samples collected with the BESIII detector at center-of-mass energies above 4.08 GeV. For the first time in this final state, a $psi(3770)$ signal is seen in the $Dbar{D}$ invariant mass spectrum, with a statistical significance of $5.2sigma$ at $sqrt{s} = 4.42$ GeV. There is also evidence for this resonance at $sqrt{s}$ = 4.26 and 4.36 GeV with statistical significance of 3.2$sigma$ and 3.3$sigma$, respectively. In addition, the Born cross section of $e^{+}e^{-}to pi^{+}pi^{-}psi(3770)$ is measured. The proposed heavy-quark-spin-symmetry partner of the $X(3872)$, the state $X_{2}(4013)$, is also searched for in the $Dbar{D}$ invariant mass spectra. No obvious signal is found. The upper limit of the Born cross section of the process $e^{+}e^{-}to rho^{0}X_{2}(4013)$ combined with the branching fraction is measured. Also, the processes $e^{+}e^{-}to D_{1}(2420)bar{D} + c.c.$ are investigated. The neutral mode with $D_{1}(2420)^{0}to D^{0}pi^{+}pi^{-}$ is reported with statistical significance of 7.4$sigma$ at $sqrt{s} = 4.42$ GeV for the first time, and evidence with statistical significance of 3.2$sigma$ and 3.3$sigma$ at $sqrt{s}$ = 4.36 and 4.60 GeV is seen, respectively. No evident signal for the process $e^{+}e^{-}to D_{1}(2420)^{0}bar{D}^{0} + c.c., D_{1}(2420)^{0}to D^{*+}pi^{-}$ is reported. Evidence for $e^{+}e^{-}to D_{1}(2420)^{+}D^{-} +~c.c., D_{1}(2420)^{+}to D^{+}pi^{+}pi^{-}$ is reported with statistical significance of 3.1$sigma$ and 3.0$sigma$ at $sqrt{s}$ = 4.36 and 4.42 GeV, respectively.
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