To investigate the interaction between the nucleon $N$ and nucleon resonance $N(1535)1/2^-$, the $eta d$ threshold structure, connected to the isoscalar $S$-wave $N$-$N(1535)1/2^-$ system, has been experimentally studied in the $gamma{d}${$to$}$pi^0eta{d}$ reaction at incident photon energies ranging from the reaction threshold to 1.15 GeV. A strong enhancement is observed near the $eta d$ threshold over the three-body phase-space contribution in the $eta d$ invariant-mass distribution. An analysis incorporating the known isovector resonance $mathcal{D}_{12}$ with a spin-parity of $2^+$ in the $pi^0d$ channel has revealed the existence of a narrow isoscalar resonance-like structure $mathcal{D}_{eta d}$ with $1^-$ in the $eta d$ system. The Breit-Wigner mass of $mathcal{D}_{eta d}$ is found to be $2.425^{-0.006}_{+0.003}$ GeV, located just close to the $eta d$ threshold, and the width is $0.034_{-0.003}^{+0.004}$ GeV. The $mathcal{D}_{eta d}$ would be a predicted isoscalar $1^-$ $eta NN$ bound state from the $eta NN$ and $pi NN$ coupled-channel calculation, or $eta d$ virtual state owing to strong $eta d$ attraction.
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