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Using 9.9 fb$^{-1}$ of $e^+e^-$ collision data collected by the BESIIIdetector at center-of-mass energies between 4.15 and 4.30 GeV, we search forthe processes $e^+e^-\to\gamma X(3872)$ with $X(3872)\rightarrow\pi^0\chi_{c0}$and $X(3872)\rightarrow\pi\pi\chi_{c0}$. Depending on the fitting model, thestatistical significance for $X(3872)\to\pi^0\chi_{c0}$ ranges from 1.3$\sigma$to 2.8$\sigma$. We set upper limits (at 90\% C.L.) of$\frac{\mathcal{B}(X(3872)\rightarrow\pi^0\chi_{c0})}{\mathcal{B}(X(3872)\to\pi^+\pi^-J/\psi)}<3.6$,$\frac{\mathcal{B}(X(3872)\rightarrow\pi^+\pi^-\chi_{c0})}{\mathcal{B}(X(3872)\to\pi^+\pi^-J/\psi)}<0.68$,and$\frac{\mathcal{B}(X(3872)\rightarrow\pi^0\pi^0\chi_{c0})}{\mathcal{B}(X(3872)\to\pi^+\pi^-J/\psi)}<1.7$.Combined with the BESIII measurement of $X(3872)\to\pi^0\chi_{c1}$, we also setan upper limit of$\frac{\mathcal{B}(X(3872)\rightarrow\pi^0\chi_{c0})}{\mathcal{B}(X(3872)\to\pi^0\chi_{c1})}<4.4$.

Search for $X(3872)\to\pi^0\chi_{c0}$ and $X(3872)\to\pi\pi\chi_{c0}$ at BESIII