  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.
