  We solve several open problems concerning integer points of polytopes arising
in symplectic and algebraic geometry. In this direction we give the first proof
of a broad case of Ewald's Conjecture (1988) concerning symmetric integral
points of monotone lattice polytopes in arbitrary dimension. We also include an
asymptotic quantitative study of the set of points appearing in Ewald's
Conjecture. Then we relate this work to the problem of displaceability of
orbits in symplectic toric geometry. We conclude with a proof for the
$2$-dimensional case, and for a number of cases in higher dimensions, of Nill's
Conjecture (2009), which is a generalization of Ewald's conjecture to smooth
lattice polytopes. Along the way the paper introduces two new classes of
polytopes which arise naturally in the study of Ewald's Conjecture and
symplectic displaceability: neat polytopes, which are related to Oda's
Conjecture, and deeply monotone polytopes.
