$w_{\infty}$-Currents in 3-Dimensional Toda Theory

Chiral densities obeying a $w_{\infty}$ Poisson--bracket algebra are constructed for the $2+1\,\, A_{\infty}$ -- Toda field theory, using its alternative $w_{\infty}$ -- Toda representation. They are obtained from formal traces of powers of the Lax operator. The spin 2 and 3 currents are explicitely derived, and the consistency of their Poisson algebra is checked.