We show that the Zamolodchikov's and Polyakov-Bershadsky nonlinear algebras $W_3$ and $W_3^{ (2)}$ can be embedded as subalgebras into some { \em linear} algebras with finite set of currents. Using these linear algebras we find new field realizations of $W_3^{ (2)}$ and $W_3$ which could be a starting point for constructing new versions of $W$-string theories. We also reveal a number of hidden relationships between $W_3$ and $W_3^{ (2)}$. We conjecture that similar linear algebras can exist for other $W$-algebras as well.
Categories: hep-th
