= Solution
If $F$ is any other <fundamental solution>, then $H=F-E$ satisfies $\partial_{\bar z}H=0$ as a <distribution>. Applying $4\partial_z$ gives $\Delta H=0$. <Elliptic regularity> for the <Laplacian> implies that the initially arbitrary <distribution> $H$ is a <smooth function>. Its equation is then the ordinary <Cauchy-Riemann equation>, so $H$ is an <entire function>.
For a direct justification of the needed regularity, choose a radial <mollifier> $\eta_r$. Locally, $H*\eta_\varepsilon$ is a smooth <harmonic function>. On a ball with sufficient room for both radii, the <mean value property> gives $(H*\eta_\varepsilon)*\eta_r=H*\eta_\varepsilon$. Passing to the <weak convergence of distributions> limit $\varepsilon\to0$ yields $H*\eta_r=H$ on a smaller ball. The left side is smooth, proving the claim locally. This establishes the <distributional Cauchy-Riemann lemma> without assuming in advance that $H$ is a <locally integrable function>.
Conversely every <entire function> $h$ satisfies $\partial_{\bar z}h=0$ and can be added to $E$. Hence all <fundamental solutions> are
$$
\boxed{F(z)=\frac1{\pi z}+h(z),\qquad h\text{ any entire function}.}
$$
No growth condition at infinity was imposed, so $h$ need not be a <polynomial>.
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