Given an ellipse with semi-major axis A, semi-minor axis B and the distance between the foci 2C, to construct an ellipse with integer values of A, B, and C that has at least one point (besides the vertexes) that has integer values for X and Y.
For the red triangle below that represents A, B, and C, any perfect triple will yield integer values. So if I set up the red triangle in terms of code numbers c^2-d^2, 2cd, c^2+d^2:
A = c^2 + d^2
B = 2*c*d
C = c^2 -d^2
The green triangle can be represented by the code number f^2-g^2, 2fg, f^2+g^2
I know that 2C is the length of the green triangle along the axis of the ellipse, so
2fg = 2(2cd)
g = 2cd/f
I also know that 2A is the sum of two sides of the green triangle:
(f^2-g^2)+(f^2+g^2) = 2(c^2+d^2)
2f^2 = 2(c^2+d^2)
c^2 + d^2 = f^2
But that's just a triple, isn't it? So as long as I choose c, d, and f so that they form a perfect triple, I can get an ellipse with integer values that has one set of integer points somewhere besides the vertexes. You need to scale the two triangles to remove the fraction caused by g.
So for c,d, and f of 3,4,5 I got an ellipse A=625, B=175, C=600 and the point (600, 49).