  It is well known that the dynamical system determined by a
Quispel-Roberts-Thompson map (a QRT map) preserves a pencil of biquadratic
polynomial curves on ${\mathbb{CP}}^1 \times {\mathbb{CP}}^1$. In most cases
this pencil is elliptic, i.e. its generic member is a smooth algebraic curve of
genus one, and the system can be solved as a translation on the elliptic fiber
to which the initial point belongs. However, this procedure is rather
complicated to handle, especially in the normalization process. In this paper,
for a given initial point on an invariant elliptic curve, we present a method
to construct the solution directly in terms of the Weierstrass sigma function,
using Kajiwara-Noumi-Yamada's parametric representation of elliptic curves.
