  We show that there is a series of topological string theories whose
integrable structure is described by the Toda lattice hierarchy. The monodromy
group of the Frobenius manifold for the matter sector is an extension of the
affine Weyl group $\widetilde W (A_N^{(1)})$ introduced by Dubrovin. These
models are generalizations of the topological $CP^1$ string theory with scaling
violation. The logarithmic Hamiltonians generate flows for the puncture
operator and its descendants. We derive the string equation from the
constraints on the Lax and the Orlov operators. The constraints are of
different type from those for the $c=1$ string theory. Higher genus expansion
is obtained by considering the Lax operator in matrix form.
