  The coupled Kadomtsev--Petviashvili system associated with an elliptic curve,
proposed by Date, Jimbo and Miwa [J. Phys. Soc. Jpn., 52:766--771, 1983], is
reinvestigated within the direct linearisation framework, which provides us
with more insights into the integrability of this elliptic model from the
perspective of a general linear integral equation. As a result, we successfully
construct for the elliptic coupled Kadomtsev--Petviashvili system not only a
Lax pair composed of differential operators in $2\times2$ matrix form but also
multi-soliton solutions with phases parametrised by points on the elliptic
curve. Dimensional reductions based on the direct linearisation, to the
elliptic coupled Korteweg-de Vries and Boussinesq systems, are also discussed.
In addition, a novel class of solutions are obtained for the $D_\infty$-type
Kadomtsev--Petviashvili equation with nonzero constant background as a
byproduct.
