How can I find all binary matrices $A$ such that $A^3$ is a non-negative, integer square matrix and

$$diag\left(A^3\right)=b$$

for some given vector $b$? Is there a way to characterize all the solutions?

How can I find all binary matrices $A$ such that $A^3$ is a non-negative, integer square matrix and

$$\mbox{diag}\left(A^3\right)=b$$

for some given vector $b$? Is there a way to characterize all the solutions?

How can I find all binary matrices $A$ that are a solution of matrix equation

$$A^3=B$$

for some given positive, integer square matrix $B$? Is there a way to characterize all the solutions?

How can I find all binary matrices $A$ such that $A^3$ is a non-negative, integer square matrix and

$$diag\left(A^3\right)=b$$

for some given vector $b$? Is there a way to characterize all the solutions?

How can I find all the binary matrices $A$ that are a solution to an equation  $$A^3=B$$ for some given, positive-integer matrix $B$? Is there a way to characterize all the solution to this problem?

How can I find all binary matrices $A$ that are a solution of matrix equation 

$$A^3=B$$

for some given positive, integer square matrix $B$? Is there a way to characterize all the solutions?