  A recently introduced model of dually weighted planar graphs is solved in
terms of an elliptic parametrization for some particular collection of planar
graphs describing the 2D $R^2$ quantum gravity. Along with the cosmological
constant $\lambda$ one has a coupling $\beta$ in the model corresponding to the
$R^2$-coupling constant. It is shown that for any value of $\beta$ the large
scale behavior of the model corresponds to that of the standard pure 2D quantum
gravity. On small distances it describes the dynamics of point-like curvature
defects introduced into the flat 2D space. The scaling function in the vicinity
of almost flat metric is obtained. The major steps of the exact solution are
given.
