  By a commutative action on a smooth quadric $Q_n$ in $P^{n+1}$ we mean an
effective action of a commutative connected algebraic group on $Q_n$ with an
open orbit. We show that for $n \geq 3$ all commutative actions on $Q_n$ are
additive actions described by Sharoiko in 2009. So there is a unique
commutative action on $Q_n$ up to equivalence. For $n = 2$ there are three
commutative actions on $Q_2$ up to equivalence, for $n = 1$ there are two
commutative actions on $Q_1$ up to equivalence.
