Find whether a dependency exist in a system of equations

I have the following system of 6 equations (all values of X and Z are constants)

 syms a1 a2 a3 b1 b2 b3

 b1-b2*b3/a1==X1
 a2-b3^2/a1==X2   
 a3-b2^2/a1==X3   
 a1-b2^2/a3==Z1   
 a2-b1^2/a3==Z2  
 b3-b1*b2/a3==Z3

I know that there is a dependency between these 6 eqs. through the following relation:

(Z3^2/Z1)-Z2=(X1^2/X3)-X2

I have two questions:

1- Is there any command in Matlab by which I can find the dependency relation between a system of equations (or can find the extra equation that if I omit, the system continue working without any loose of information)?

2- There are 6 eqs. in above system. However, because of dependency, in fact I have 5 eqs. and 6 unknowns. So, if I try to solve, it should return infinity number of answers. However, in many cases "solve" gives me only 3 answers for that. What is the reason?

(e.g. for X1=2/3, X2=5/3, X3=2/3, Z1=2, Z2=1, Z3=0)