  Based on $(10.09 \pm 0.04) \times 10^9$ $J/\psi$ events collected with the
BESIII detector operating at the BEPCII collider, a partial wave analysis of
the decay $J/\psi \to \phi \pi^{0}\eta$ is performed. We observe for the first
time two new structures on the $\phi\eta$ invariant mass distribution, with
statistical significances of $24.0\sigma$ and $16.9\sigma$; the first with
$J^{\rm PC}$ = $1^{+-}$, mass M = (1911 $\pm$ 6 (stat.) $\pm$ 14
(sys.))~MeV/$c^{2}$, and width $\Gamma = $ (149 $\pm$ 12 (stat.) $\pm$ 23
(sys.))~MeV, the second with $J^{\rm PC}$ = $1^{--}$, mass M = (1996 $\pm$ 11
(stat.) $\pm$ 30 (sys.))~MeV/$c^{2}$, and width $\Gamma$ = (148 $\pm$ 16
(stat.) $\pm$ 66 (sys.))~MeV. These measurements provide important input for
the strangeonium spectrum. In addition, the $f_0(980)-a_0(980)^0$ mixing signal
in $J/\psi \to \phi f_0(980) \to \phi a_0(980)^0$ and the corresponding
electromagnetic decay $J/\psi \to \phi a_0(980)^0$ are measured with improved
precision, providing crucial information to understand the nature of
$a_0(980)^0$ and $f_0(980)$.
