  Fix an algebraically closed field $\F$ and an integer $d \geq 3$. Let $V$ be
a vector space over $\F$ with dimension $d+1$. A Leonard pair on $V$ is a pair
of diagonalizable linear transformations $A: V \to V$ and $A^* : V \to V$, each
acting in an irreducible tridiagonal fashion on an eigenbasis for the other
one. There is an object related to a Leonard pair called a Leonard system. It
is known that a Leonard system is determined up to isomorphism by a sequence of
scalars $(\{\th_i\}_{i=0}^d, \{\th^*_i\}_{i=0}^d, \{\vphi_i\}_{i=1}^d,
\{\phi_i\}_{i=1}^d)$, called its parameter array. The scalars
$\{\th_i\}_{i=0}^d$ (resp.\ $\{\th^*_i\}_{i=0}^d$) are mutually distinct, and
the expressions $(\th_{i-2} - \th_{i+1})/(\th_{i-1}-\th_{i})$, $(\th^*_{i-2} -
\th^*_{i+1})/(\th^*_{i-1}-\th^*_{i})$ are equal and independent of $i$ for $2
\leq i \leq d-1$. Write this common value as $\beta+1$. In the present paper,
we consider the "end-parameters" $\th_0$, $\th_d$, $\th^*_0$, $\th^*_d$,
$\vphi_1$, $\vphi_d$, $\phi_1$, $\phi_d$ of the parameter array. We show that a
Leonard system is determined up to isomorphism by the end-parameters and
$\beta$. We display a relation between the end-parameters and $\beta$. Using
this relation, we show that there are up to inverse at most $\lfloor (d-1)/2
\rfloor$ Leonard systems that have specified end-parameters. The upper bound
$\lfloor (d-1)/2 \rfloor$ is best possible.
