= Solution
Let $\omega$ solve the equation represented by the original symbol and define a <dependent-variable rescaling of a Papperitz symbol>
$$
\omega_1(z)=\left(\frac{z-a}{z-b}\right)^\delta\omega(z),
$$
with a consistent local choice of branch. Near $a$, the multiplier is a nonzero analytic factor times $(z-a)^\delta$, so both exponents at $a$ increase by $\delta$. Near $b$, it is a nonzero analytic factor times $(z-b)^{-\delta}$, so both exponents at $b$ decrease by $\delta$. It is analytic and nonzero near $c$, so the exponents there are unchanged. Therefore the transformed equation has symbol
$$
\boxed{
P\left\{
\begin{matrix}
a&b&c\\
\alpha+\delta&\beta-\delta&\gamma\\
\alpha'+\delta&\beta'-\delta&\gamma'
\end{matrix};z
\right\}}.
$$
The opposite shifts also preserve the <Fuchs relation>.
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