@Carl Love 

Updated. It is a square. 

@acer 

Thank you a lot. That is what I am looking for. Unfortunately, I am using the older Maple version. I can not run the "CP" function. Anyway, I appreciate your kind help very much!

@acer 

Sorry for the unclear of the question. I have updated the question. Thank you for your help always. 

@acer 

I see. It works. Thank you very much again for your kind help!

@mmcdara 

Thank you for your reply. I am sorry that perhaps I didn't make my question clear. 

I want to plot the 2d contour (x-y plane) in which the color of the line of the 2d contour is same as that in the 3d plot.

Thank you a lot for your help!

@Kitonum & @Carl Love 

Thank you for both of your help very much!

@acer 

I found the label after the transformation is disappeared as shown as follows. How do I solve it? Thank you.

The codes are as follows:

****************************************************

with(plots);

Eqn01:=sqrt((1+48/(-8*X^2+16*X*t-24*t^2-12))^2+1024*X^2/(-8*X^2+16*X*t-24*t^2-12)^2);

x:=X+t+(16*(-2*t*(-1-I*sqrt(2))*(-1+I*sqrt(2))-X*(-1-I*sqrt(2))-X*(-1+I*sqrt(2))))/(-(4*(-1-I*sqrt(2)))*(-1+I*sqrt(2))-(8*(-t*(-1+I*sqrt(2))-X))*(-t*(-1-I*sqrt(2))-X));

ContoursWithLabels:= proc(

     Expr::algebraic,

     Range1::(name= range(realcons)), Range2::(name= range(realcons)),

     {contours::{posint, {set,list}(realcons)}:= 8},

     {ImplicitplotOptions::{list,set}({name, name= anything}):= NULL},

     {GraphicOptions::{list,set}({name, name= anything}):= NULL},

     {TextOptions::{list,set}({name, name= anything}):= NULL},

     {Coloring::{list,set}({name, name= anything}):= NULL}

)

local

     r1, r2, f, L1, h, S1, P, r, M, C, T, p, p1, m, n, i,

     x:= lhs(Range1), y:= lhs(Range2)

;

     f:= unapply(Expr, (x,y));

     if contours::posint then

          r1:= rand(convert(rhs(Range1), float));

          r2:= rand(convert(rhs(Range2), float));

          L1:= select(type, (f@op)~({seq([r1,r2](), i= 1..205)}), realcons);

          h:= (L1[-6]-L1[1])/contours;

          S1:= [seq(L1[1]+h/2+h*(n-1), n= 1..contours)]

     else #contours::{set,list}(realcons)

          S1:= [contours[]]

     end if;

     userinfo(1, ContoursWithLabels, print('Contours' = evalf[2](S1)), `\n`);

     r:= k-> rand(20..k-20);

     for C in S1 do

          P:= plots:-implicitplot(

               Expr = C, Range1, Range2,

               gridrefine= 3, ImplicitplotOptions[]

          );

          for p in [plottools:-getdata(P)] do

               p1:= convert(p[3], listlist);

               n:= nops(p1);

               if n < 500 then

                    m:= `if`(40 < n, r(n)(), round(n/2));

                    M[`if`(40 < n, [p1[1..m-11], p1[m+11..n]], [p1])[]]:= NULL;

                    T[[p1[m][], evalf[2](C)]]:= NULL

               else

                    h:= trunc(n/2);

                    m:= r(h)();

                    M[p1[1..m-11], p1[m+11..m+h-11], p1[m+h+11..n]]:= NULL;

                    T[[p1[m][], evalf[2](C)], [p1[m+h][], evalf[2](C)]]:= NULL

               end if

          end do

     end do;

     plots:-display(

          [`if`(

               Coloring = NULL,

               NULL,

               plots:-densityplot(Expr, Range1, Range2, Coloring[])

          ),

          plot([indices(M, 'nolist')], color= black, GraphicOptions[]),

          plots:-textplot([indices(T, 'nolist')], TextOptions[])

         ], 'axes'= 'box', 'gridlines'= false, _rest

     )

end proc:

conts1:=[0.1, 0.3, 0.5, 0.7, 0.9, 1.3, 2.0];

Pft := ContoursWithLabels(

       Eqn01,X=-20..20, t=-20..20, contours= conts1,

       TextOptions=[font=[HELVETICA,BOLD,10], color= black],

       ImplicitplotOptions=[gridrefine=4],

       GraphicOptions=[thickness=0],

       labelfont=[TIMES,BOLDITALIC,10], axesfont=[HELVETICA,18]);

P2D1 :=plots:-display(

  plottools:-transform((a,b)->eval([x,t],[ X =a,t=b]))(Pft),

  labels=["x","t"]

);

@tomleslie 

Thank you for your help. It works.

@acer 

Thank you very much! I appreciate your kind help.

@acer  Thank you very much for your help !

Can I project this two-dimensional contour plot (graph of x-t) on the bottom of a three dimensional plot (graph of u versus x-t)? i.e. a graph shows both the topology and the countours. Thank you again.

@acer It works as well and it is nice.

I appreciate both of your kind help (@tomleslie & @acer) very much.

@acer 

Thank you for your reply. I want to plot all the imaginary part of z as s varies for whatever the real part of z.

@tomleslie 

Thank you very much for your help!!