We consider the problem of reallocating the total initial endowments o
f an infinitely divisible commodity among agents with single-peaked pr
eferences. With the uniform reallocation rule we propose a solution wh
ich satisfies many appealing properties, describing the effect of popu
lation and endowment variations on the outcome. The central properties
which are studied in this context are population monotonicity, bilate
ral consistency, (endowment) monotonicity and (endowment) strategy-pro
ofness. Furthermore, the uniform reallocation rule is Pareto optimal a
nd satisfies several equity conditions, e.g., equal-treatment and envy
-freeness. We study the trade-off between properties concerning variat
ion and properties concerning equity. Furthermore, we provide several
characterizations of the uniform reallocation rule based on these prop
erties.