Two series of W-algebras with two generators are constructed from chiral vertex operators of a free field representation. If $c = 1 - 24k$, there exists a W(2,3k) algebra for k in $Z_{ +}/2$ and a W(2,8k) algebra for k in $Z_{ +}/4$. All possible lowest-weight representations, their characters and fusion rules are calculated proving that these theories are rational. It is shown, that these non-unitary theories complete the classification of all rational theories with effective central charge $c_{ eff} = 1$. The results are generalized to the case of extended supersymmetric conformal algebras.
Categories: hep-th