  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 $D\bar{D}$ invariant mass
spectrum, with a statistical significance of $5.2\sigma$ 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 $D\bar{D}$ 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.
