  Given a rigid realisation of a graph $G$ in ${\mathbb R}^2$, it is an open
problem to determine the maximum number of pairwise non-congruent realisations
which have the same edge lengths as the given realisation. This problem can be
restated as finding the number of solutions of a related system of quadratic
equations and in this context it is natural to consider the number of solutions
in ${\mathbb C}^2$ rather that ${\mathbb R}^2$. We show that the number of
complex solutions, $c(G)$, is the same for all generic realisations of a rigid
graph $G$, characterise the graphs $G$ for which $c(G)=1$, and show that the
problem of determining $c(G)$ can be reduced to the case when $G$ is
$3$-connected and has no non-trivial $3$-edge-cuts. We consider the effect of
the Henneberg moves and the vertex-splitting operation on $c(G)$. We use our
results to determine $c(G)$ exactly for two important families of graphs, and
show that the graphs in both families have $c(G)$ pairwise equivalent generic
real realisations. We also show that every planar isostatic graph on $n$
vertices has at least $2^{n-3}$ pairwise equivalent real realisations.
