  Chas and Sullivan introduced string homology, which is the equivariant
homology of the loop space with the $S^1$ action on loops by rotation. Craig
Westerland computed the string homology for spheres with coefficients in
$\mathbb{Z} /2\mathbb{Z}$ and Somnath Basu computed the string homology and
string bracket for spheres over rational coefficients and found that the
bracket is trivial in his dissertation. In this paper, we compute string
homology and the string bracket for spheres with integer coefficients, treating
the odd- and even-dimensional cases separately. We use the Gysin sequence and
Leray-Serre spectral sequence for our computations. We find that over the
integers, the string Lie algebra bracket structure is not always zero as Basu
found. The string bracket turns out to be non-zero on torsion elements coming
from string homology.
