Several inverse spectral problems are solved by a method which is
based on exact solutions of the semi-infinite Toda lattice. In
fact, starting with a well-known and appropriate probability
measure $\mu$ , the solution $\alpha_n(t)$ , $b_n(t)$ of
the Toda lattice is exactly determined and by taking $t=0$ ,
the solution $\alpha_n(0)$ , $b_n(0)$ of the inverse
spectral problem is obtained. The solutions of the Toda lattice
which are found in this way are finite for every $t>0$ and can
also be obtained from the solutions of a simple differential
equation. Many other exact solutions obtained from this
differential equation show that there exist initial conditions
$\alpha_n(0)>0$ and $b_n(0)\in \mathbb{R}$ such that the
semi-infinite Toda lattice is not integrable in the sense that
the functions $\alpha_n(t)$ and $b_n(t)$ are not finite for every
$t>0$ .