  We construct a generalization of the cyclic $\lambda$-deformed models of
\cite{Georgiou:2017oly} by relaxing the requirement that all the WZW models
should have the same level $k$. Our theories are integrable and flow from a
single UV point to different IR fixed points depending on the different
orderings of the WZW levels $k_i$. First we calculate the Zamolodchikov's
C-function for these models as exact functions of the deformation parameters.
Subsequently, we fully characterize each of the IR conformal field theories.
Although the corresponding left and right sectors have different symmetries,
realized as products of current and coset-type symmetries, the associated
central charges are precisely equal, in agreement with the valuesobtained from
the C-function.
