= Solution
Normalize the second-order equation as $y''+P(z)y'+Q(z)y=0$, with $P,Q$ holomorphic at all finite nonzero points. The <regular singular point criterion for a second-order equation> at zero says that $zP(z)$ and $z^2Q(z)$ extend holomorphically across zero.
To impose the condition at infinity, set $w=1/z$ and $Y(w)=y(1/w)$. Since $y'=-w^2Y'$ and $y''=w^4Y''+2w^3Y'$, the transformed equation is
$$
Y''+\left(\frac2w-\frac{P(1/w)}{w^2}\right)Y'+\frac{Q(1/w)}{w^4}Y=0.
$$
Regular singularity at $w=0$ makes $2-P(1/w)/w$ and $Q(1/w)/w^2$ bounded and holomorphic there. Equivalently $zP(z)$ and $z^2Q(z)$ are bounded at infinity. They are entire functions by the zero-endpoint condition, so <Liouville's theorem> makes both constants. The <Euler-Cauchy classification from two regular singular points> is therefore
$$
\boxed{z^2y''+Azy'+By=0,\qquad A,B\in\mathbb C.}
$$
Conversely these equations have ordinary coefficients everywhere on $\mathbb C^*$ and meet both regular-singularity bounds. If “regular singular” is required to exclude an ordinary point, omit $(A,B)=(0,0)$, which makes zero ordinary, and $(2,0)$, which makes infinity ordinary. Under the common “at worst regular singular” convention these two cases are included. Multiplication by a nonvanishing coefficient does not change the normalized equation or this classification.
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