mathematics - Does this mathy square have any solutions? (And how many?)

Consider a $5\times5$ grid of math operators and numbers that encodes 8 math equations:

A + B = C
= + + + +
D + E = F
+ = = = =
G = H + I

There are 3 horizontal equations, 3 vertical, and 2 diagonal. Specifically:

A + B = C
D + E = F
H + I = G
D + G = A
B + E = H
C + F = I
A + E = I
C + E = G

However, to make the grid less strict, each equation has 3 additional allowed forms:

• 
  

  The + can be replaced with *. e.g. A * B = C. (We'll avoid - / ^ since they aren't commutative.)

  
  


• 
  

  The + and = can be swapped. e.g. A = B + C.

  
  


• 
  
  

  Both of the above. e.g. A = B * C.

  
  

Thus you can massage the equations a bit in order to form a more workable grid. For example, it might look like:

A * B = C
+ = * * +
D = E * F
= = = + =
G * H = I

The question is: Can the letters A through I be replaced with all one-digit numbers 1 through 9 in any order such that all 8 grid equations are satisfied?

Are there lots of solutions or none at all?

(Bonus: What if - or / or ^ is allowed to be an operator (and you make some assumptions about reading direction).)