  We show how to construct semi-invariants and integrals of the full symmetric
sl(n) Toda lattice for all n. Using the Toda equations for the Lax eigenvector
matrix we prove the existence of semi-invariants which are homogeneous
coordinates in the corresponding projective spaces. Then we use these
semi-invariants to construct the integrals. The existence of additional
integrals which constitute a full set of independent non-involutive integrals
was known but the chopping and Kostant procedures have crucial computational
complexities already for low-rank Lax matrices and are practically not
applicable for higher ranks. Our new approach solves this problem and results
in simple explicit formulae for the full set of independent semi-invariants and
integrals expressed in terms of the Lax matrix and its eigenvectors, and of
eigenvalue matrices for the full symmetric sl(n) Toda lattice.
