p1 := plots[animate](plot3d, [[y, x, (1/3)*Pi], x = 0 .. 2*Pi, y = 0 .. R, coords = spherical, scaling = constrained], R = 0 .. 5):
p2 := plots[animate](plot3d, [[y, x, (1/3)*Pi], x = 0 .. p, y = 0 .. 5, coords = spherical, scaling = constrained], p = 0 .. 2*Pi):
plots[display]([p1, p2], axes = normal);

...

solve({f(x,y)});

assign(%):  # right after it !!

   ...

unassign('x','y'):  # or use "restart" instead 

Unfortunely.We can't, becase Maple treat many math commands in a default assuming which we don't usually needed or expected.

But there is a "hard" way to suit  what you expected,using Maple in a "Basic Command" way. The attached picture is an example for your reference, yet it fails to suit your needs since Maple always treat (x+1)/2 as x/2+1/2,and that is what I mentoned, too many pedefined assumptions that make user confused.