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By using the interplay between the Eulerian idempotent and the Dynkin idempotent, we construct explicitly a particular symmetric solution $(F,G)$ of the first equation of the Kashiwara-Vergne conjecture $$x+y-log\left({\mathrm{e}}^y{\mathrm{e}}^x\right)=(1-{\mathrm{e}}^{-\mathrm{ad}x})F(x,y)+({\mathrm{e}}^{\mathrm{ad}y}-1)G(x,y).$$ Then, we explicit all the solutions of the equation in the completion of the free Lie algebra generated by two indeterminates $x$ and $y$ thanks to the kernel of the Dynkin idempotent.