I am sorry, yes of course there is fsolve.  Here are my explicit equations.

x := 2* (-sin(2* a) + sin(2* (a + b)) +
     2* cos(2* a + 2* b + 2* c + d) *sin(d));
 
 y := -2 + 4*cos(2* a) - 4*cos(2* (a + b)) +
   4*cos(2* (a + b + c)) - 4*cos(2* (a + b + c + d));
 
 z := 2* (sin(a) - sin(a + b) + sin(a + b + c) -
     sin(a + b + c + d));
 
 w := -1 + 2* cos(a) - 2* cos(a + b) + 2* cos(a + b + c) -
   2* cos(a + b + c + d);
 
 S1:=fsolve({x=0,y=0,z=0,w=0},{a=1.0,b=1.0,c=1.0,d=1.0},{a=0..7,b=0..7,c=0..7,d=0..7});

And then this gives the same result:

S2:=fsolve({x=0,y=0,z=0,w=0},{a=1.0,b=1.0,c=1.0,d=1.0},{a=0..7,b=0..7,c=0..7,d=0..7},avoid=S1);

Any ideas?