  We study a biased $2\times 2$ periodic random domino tilings of the Aztec
diamond and associate a linear flow on an elliptic curve to this model. Our
main result is a double integral formula for the correlation kernel, in which
the integrand is expressed in terms of this flow. For special choices of
parameters the flow is periodic, and this allows us to perform a saddle point
analysis for the correlation kernel. In these cases we compute the local
correlations in the smooth disordered (or gaseous) region. The special example
in which the flow has period six is worked out in more detail, and we show that
in that case the boundary of the rough disordered region is an algebraic curve
of degree eight.
