  Consider the celebrated Lyness recurrence $x_{n+2}=(a+x_{n+1})/x_{n}$ with
$a\in\Q$. First we prove that there exist initial conditions and values of $a$
for which it generates periodic sequences of rational numbers with prime
periods $1,2,3,5,6,7,8,9,10$ or $12$ and that these are the only periods that
rational sequences $\{x_n\}_n$ can have. It is known that if we restrict our
attention to positive rational values of $a$ and positive rational initial
conditions the only possible periods are $1,5$ and $9$. Moreover 1-periodic and
5-periodic sequences are easily obtained. We prove that for infinitely many
positive values of $a,$ positive 9-period rational sequences occur. This last
result is our main contribution and answers an open question left in previous
works of Bastien \& Rogalski and Zeeman. We also prove that the level sets of
the invariant associated to the Lyness map is a two-parameter family of
elliptic curves that is a universal family of the elliptic curves with a point
of order $n, n\ge5,$ including $n$ infinity. This fact implies that the Lyness
map is a universal normal form for most birrational maps on elliptic curves.
