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 Zamolodchikovs 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.
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