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Based on (10.09±0.04)×109 J/ψ events collected with the BESIII detector operating at the BEPCII collider, a partial wave analysis of the decay J/ψ→ϕπ0η is performed. We observe for the first time two new structures on the ϕη invariant mass distribution, with statistical significances of 24.0σ and 16.9σ; the first with JPC = 1+−, mass M = (1911 ± 6 (stat.) ± 14 (sys.))~MeV/c2, and width Γ= (149 ± 12 (stat.) ± 23 (sys.))~MeV, the second with JPC = 1−−, mass M = (1996 ± 11 (stat.) ± 30 (sys.))~MeV/c2, and width Γ = (148 ± 16 (stat.) ± 66 (sys.))~MeV. These measurements provide important input for the strangeonium spectrum. In addition, the f0(980)−a0(980)0 mixing signal in J/ψ→ϕf0(980)→ϕa0(980)0 and the corresponding electromagnetic decay J/ψ→ϕa0(980)0 are measured with improved precision, providing crucial information to understand the nature of a0(980)0 and f0(980).