  We consider the problem of the real analytic dependence of the accessory
parameters of Liouville theory on the moduli of the problem, for general
elliptic singularities. We give a simplified proof of the almost everywhere
real analyticity in the case of a single accessory parameter as it occurs e.g.
in the sphere topology with four sources or for the torus topology with a
single source by using only the general analyticity properties of the solution
of the auxiliary equation. We deal then the case of two accessory parameters.
We use the obtained result for a single accessory parameter to derive rigorous
properties of the projection of the problem on lower dimensional planes. We
derive the real analyticity result for two accessory parameters under an
assumption of irreducibility.
