  The resonant substructures of $B^0 \to \overline{D}^0 \pi^+\pi^-$ decays are
studied with the Dalitz plot technique. In this study a data sample
corresponding to an integrated luminosity of 3.0 fb$^{-1}$ of $pp$ collisions
collected by the LHCb detector is used. The branching fraction of the $B^0 \to
\overline{D}^0 \pi^+\pi^-$ decay in the region $m(\overline{D}^0\pi^{\pm})>2.1$
GeV$/c^2$ is measured to be $(8.46 \pm 0.14 \pm 0.29 \pm 0.40) \times 10^{-4}$,
where the first uncertainty is statistical, the second is systematic and the
last arises from the normalisation channel $B^0 \to D^*(2010)^-\pi^+$. The
$\pi^+\pi^-$ S-wave components are modelled with the Isobar and K-matrix
formalisms. Results of the Dalitz plot analyses using both models are
presented. A resonant structure at $m(\overline{D}^0\pi^-) \approx 2.8$
GeV$/c^{2}$ is confirmed and its spin-parity is determined for the first time
as $J^P = 3^-$. The branching fraction, mass and width of this structure are
determined together with those of the $D^*_0(2400)^-$ and $D^*_2(2460)^-$
resonances. The branching fractions of other $B^0 \to \overline{D}^0 h^0$ decay
components with $h^0 \to \pi^+\pi^-$ are also reported. Many of these branching
fraction measurements are the most precise to date. The first observation of
the decays $B^0 \to \overline{D}^0 f_0(500)$, $B^0 \to \overline{D}^0
f_0(980)$, $B^0 \to \overline{D}^0 \rho(1450)$, $B^0 \to D_3^*(2760)^- \pi^+$
and the first evidence of $B^0 \to \overline{D}^0 f_0(2020)$ are presented.
