Integrals of open 2D lattices

We present an explicit formula for integrals of the open 2D Toda lattice of type $A_n$. This formula is applicable for various reductions of this lattice. To illustrate the concept we find integrals of the Toda $G_2$ lattice. We also reveal a connection between the open Toda $A_n$ and Shabat-Yamilov lattices.