Searches for isospin-violating transitions<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi>χ</mml:mi><mml:mrow><mml:mi>c</mml:mi><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">→</mml:mo><mml:msup><mml:mi>π</mml:mi><mml:mn>0</mml:mn></mml:msup><mml:msub><mml:mi>η</mml:mi><mml:mi>c</mml:mi></mml:msub></mml:math>

We present the first upper-limit measurement of the branching fractions of the isospin-violating transitions ${\ensuremath{\chi}}_{c0,2}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}{\ensuremath{\eta}}_{c}$. The measurements are performed using $106\ifmmode\times\else\texttimes\fi{}1{0}^{6}$ $\ensuremath{\psi}(3686)$ events accumulated with the BESIII detector at the BEPCII ${e}^{+}{e}^{\ensuremath{-}}$ collider at a center-of-mass energy corresponding to the $\ensuremath{\psi}(3686)$ mass. We obtained upper limits on the branching fractions at a 90% confidence level of $B({\ensuremath{\chi}}_{c0}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}{\ensuremath{\eta}}_{c})&lt;1.6\ifmmode\times\else\texttimes\fi{}1{0}^{\ensuremath{-}3}$ and $B({\ensuremath{\chi}}_{c2}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}{\ensuremath{\eta}}_{c})&lt;3.2\ifmmode\times\else\texttimes\fi{}1{0}^{\ensuremath{-}3}$.