  We study the two-channel Kondo problem in the context of two interacting
helical liquids coupled to a spin-$\frac12$ magnetic impurity. We show that the
interactions between the two helical liquids significantly affect the phase
diagram and other observable properties. Using a multichannel Luttinger liquid
formalism, we analyze both the Toulouse limit, where an exact solution is
available, and the weak coupling limit, which can be studied via a perturbative
renormalization group (RG) approach. We recover the results for the `decoupled'
limit (interactions between the helical liquids switched off) and point out
deviations from the known results due to this coupling. The model under study
is mapped to a model of two effectively decoupled helical liquids coupled to an
impurity. The perturbative RG study shows that each of these channels can flow
to either a Ferromagnetic (FM) or an Anti-Ferromagnetic (AFM) fixed point. We
obtain the phase diagram of the coupled system as a function of the system
parameters. The observable consequences of the interaction between the two
channels are captured using linear response theory. We compute the negative
correction to the conductance due to the Kondo scattering processes and show
how it scales with the temperature as a function of inter-channel interaction.
