  In this paper we continue the study of $Q$-operators in the six-vertex model
and its higher spin generalizations. In [1] we derived a new expression for the
higher spin $R$-matrix associated with the affine quantum algebra
$U_q(\widehat{sl(2)})$. Taking a special limit in this $R$-matrix we obtained
new formulas for the $Q$-operators acting in the tensor product of
representation spaces with arbitrary complex spin.
  Here we use a different strategy and construct $Q$-operators as integral
operators with factorized kernels based on the original Baxter's method used in
the solution of the eight-vertex model. We compare this approach with the
method developed in [1] and find the explicit connection between two
constructions. We also discuss a reduction to the case of finite-dimensional
representations with (half-) integer spins.
