Given: Coordinates of two diagonally opposite vertices of a rectangle are (4, 3) and (-4,-3).

To find: The number of such rectangle(s), if the other two vertices also have integral coordinates.

Approach:
Let's consider the rectangle ABCD, where A and C are the given points (4,3) and (-4,-3) respectively.

We know that the opposite sides of a rectangle are equal and parallel to each other. Hence, the other two vertices B and D should lie on the lines passing through A and C respectively, such that AB = CD and BC = AD.

Let's find the equation of the line passing through A and C:

Slope of AC, m = (y2 - y1) / (x2 - x1)
= (-3 - 3) / (-4 - 4)
= -6/(-8)
= 3/4

Using the point-slope form of a line, we can find the equation of the line passing through A:

y - y1 = m(x - x1)
y - 3 = (3/4)(x - 4)
y = (3/4)x - 3

Similarly, we can find the equation of the line passing through C:

y + 3 = (3/4)(x + 4)
y = (3/4)x - 9/4

Now, let's find the coordinates of the other two vertices B and D such that AB = CD and BC = AD:

AB = CD
=> Distance between A and B = Distance between C and D
=> (x2 - x1)^2 + (y2 - y1)^2 = (x4 - x3)^2 + (y4 - y3)^2

Substituting the values, we get:

(x2 - (-4))^2 + (y2 - (-3))^2 = (x4 - 4)^2 + (y4 - 3)^2

Simplifying the above equation, we get:

(x2 + 4)^2 + (y2 + 3)^2 = (x4 - 4)^2 + (y4 - 3)^2

BC = AD
=> Distance between B and C = Distance between A and D
=> (x3 - x2)^2 + (y3 - y2)^2 = (x4 - x1)^2 + (y4 - y1)^2

Substituting the values, we get:

(x3 - x2)^2 + (y3 - y2)^2 = (x4 - 4)^2 + (y4 - 3)^2

(x4 - 4)^2 + (y4 - 3)^2 = (x3 - x2)^2 + (y3 - y2)^2

We need to find the integral coordinates of B and D. Let's consider the possible cases:

Case 1: B and D lie on the same side of AC

In this case, B and D can lie on the lines perpendicular to AC passing through A and C respectively. Let's find the equations of these lines:

Equation of the line perpendicular to AC passing through A:

Slope of perpendicular line, m' = -1/m
= -4/3

Using the point-slope form of a