  The classes of Monge-Amp\`ere systems, decomposable and bi-decomposable
Monge-Amp\`ere systems, including equations for improper affine spheres and
hypersurfaces of constant Gauss-Kronecker curvature are introduced. They are
studied by the clear geometric setting of Lagrangian contact structures, based
on the existence of Lagrangian pairs in contact structures. We show that the
Lagrangian pair is uniquely determined by such a bi-decomposable system up to
the order, if the number of independent variables $\geq 3$. We remark that, in
the case of three variables, each bi-decomposable system is generated by a
non-degenerate three-form in the sense of Hitchin. It is shown that several
classes of homogeneous Monge-Amp\`ere systems with Lagrangian pairs arise
naturally in various geometries. Moreover we establish the upper bounds on the
symmetry dimensions of decomposable and bi-decomposable Monge-Amp\`ere systems
respectively in terms of the geometric structure and we show that these
estimates are sharp (Proposition 4.2 and Theorem 5.3).
