  We apply the conformal method to solve the initial value formulation of
general relativity to the $\lambda$-R model, a minimal, anisotropic
modification of general relativity with a preferred foliation and two local
degrees of freedom. We obtain a generalised Lichnerowicz-York equation for the
conformal factor of the metric and derive its properties. We show that the
behaviour of the equation depends on the value of the coupling $\lambda$. In
the absence of a cosmological constant, we recover the existence and uniqueness
properties of the original equation when $\lambda>1/3$ and the trace of the
momentum of the metric, $\pi$, is non-vanishing. For $\pi=0$, we recover the
original Lichnerowicz equation regardless of the value of $\lambda$ and must
therefore restrict the metric to the positive Yamabe class. The same
restriction holds for $\lambda<1/3$, a case in which we show that the spatial
Ricci scalar must also be large enough to guarantee the existence of at least
one solution. Taking the equations of motion into account, this allows us to
prove that there is in general no way of matching both constraint solving data
and time evolution of phase space variables between the $\lambda$-R model and
general relativity, thereby proving the the non-equivalence between the
theories outside of the previously known cases $\lambda=1$ and $\pi=0$.
