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In this paper, we mainly investigate the $W_{2,s}^{M}\otimes W_{2,s}^{L}$system, in which the matter and the Liouville subsystems generate $W_{2,s}^{M}$and $W_{2,s}^L$ algebras respectively. We first give a brief discussion of thephysical states for corresponding $W$ stings. The lower states are given byfreezing the spin-2 and spin-$s$ currents. Then, introducing two pairs ofghost-like fields, we give the realizations of $W_{1,2,s}$ algebras. Based onthese linear realizations, BRST operators for $W_{2,s}$ algebras are obtained.Finally, we construct new BRST charges of Liouville system for $W_{2,s}^{L}$strings at the specific values of central charges $c$: $c=-{22/5}$ for$W_{2,3}^{L}$ algebra, $c=-24$ for $W_{2,4}^{L}$ algebra and $c=-2,-{286/3}$for $W_{2,6}^{L}$ algebra, at which the corresponding $W_{2,s}^L$ algebras aresingular.

Physical States and BRST Operators for Higher-spin $W$ Strings