Let x=a, y=b, z=c. Start with the positive integers:

1/2+1/3+1/6=1

1/3+1/3+1/3=1

1/2+1/4+1/4=1, and these are all possible solutions. 

1/2+1/3+1/6=1 does not work, because 2 and 3 are prime.

1/3+1/3+1/3=1 works for {x,y,z} = {3,1,1}

1/2+1/4+1/4=1 works for {x,y,z} = {2,2,1}

Since 1/x cannot exceed 1, its greatest possible value is 1

Assuming that is so, then (1/xy)=-(1/xyz), or z+1 = 0, so z=-1, where {x,z} = {1, -1} and y can be any nonzero integer: e.g. 1/1+ 1/(1*13)+ 1/(1*13*-1) 

The last alternative is that x is negative. If so then x cannot be less than -1, for if so the fractions 1/xy+1/xyz would be too small to satisfy the given equation. 

So the final solution is {x,y,z} = {-1,-1, 1}