@acer Yes, I think the inner shell would be an union. In fact, I need to plot the volume for different parameters of the spindle tori. So, their inner and outer shells might intersect in the same or different ways. Hence, I would prefer if I do not need to think about the union or intersection of the inner or the outer shells instead plot the volume that simply satisfies the four inequalities fi<0, i=1,2,3,4.

@acer Thank you for this method. The surface is indeed smooth.

But there is something wrong with the code that it doesn't plot all the points where fi<0, i=1,2,3,4. 

If you change the view option while plotting from [-1.7..1.7,-1.7..1.7,-1.7..1.7] to [-1.7..1.7,-1.7..1.7,0..1.7], you will see that the inner surface is missing unlike if it is plotted using the implicitplot3d.

@Kitonum Thank you very much. It worked like a charm!

I  didn't realize that the equations were posted as images. Please find them here which can be copied to Maple directly.

f1:=x^4+2*x^2*y^2+2*x^2*z^2+y^4+2*y^2*z^2+z^4-2*x^3-2*x*y^2-2*x*z^2-(79/25)*x^2-(104/25)*y^2-(8/5)*z^2+(104/25)*x;
f2:=x^4+2*x^2*y^2+2*x^2*z^2+y^4+2*y^2*z^2+z^4-2*x^2*y-2*y^3-2*y*z^2-(104/25)*x^2-(79/25)*y^2-(8/5)*z^2+(104/25)*y;
f3:=x^4+2*x^2*y^2+2*x^2*z^2+y^4+2*y^2*z^2+z^4+2*x^3+2*x*y^2+2*x*z^2-(79/25)*x^2-(104/25)*y^2-(8/5)*z^2-(104/25)*x;
f4:=x^4+2*x^2*y^2+2*x^2*z^2+y^4+2*y^2*z^2+z^4+2*x^2*y+2*y^3+2*y*z^2-(104/25)*x^2-(79/25)*y^2-(8/5)*z^2-(104/25)*y;

@Robert Israel This is perfect, thank you but what if the equations are non-linear in x,y and z?

@Carl Love I cannot use solve or eliminate as I have only one equation (It's a trigonometric equation). But I did not know about DirectSearch. I will try that one. Thanks.

@Carl Love Thank you,

I have a slightly different question than my original post. If implicitplot is not numeric, then is there any way to solve a multivariate implicit function?

For example I have a an equation f(Z, theta, phi) in theta, phi and Z. 0<theta<pi, -pi<phi<pi and 0<=Z<4. Can I find all values of Z, theta and phi for which f(Z, theta, phi)=0? Infact, I am expecting just two solution sets.