Simultaneous Equations

Consider the following three(3) equations under the given conditions.

M^1/2 = a(a^6 + Aa^4b^2 + Ba^2b^4 + Cb^6) (1)

N^1/2 = b(b^6 + Ab^4a^2 + Bb^2a^4 + Ca^6) (2)

Z = a^2 + b^2 (3)

Conditions: M, N, Z are nonsquare integers each > 1, Z is odd and (M,
N, Z) = 1
A, B, C are integers > 3. a and b are different each > 0.

Assertion: (1), (2) and (3) can simultaneously be satisfied only if a =
g^1/2 and b = h^1/2.
g and h are nonsquare integers such that (g, h) = 1, h is even.

My arguments: Since a and b contains only even powers inside ( ) in
(1) and (2) these two equations shall be consistent if a and b are
chosen in the forms as suggested above. With these chosen values of a
and b (3) can be satisfied. Is there any counter argument?
Any comments about the correctness of the assertion will be appreciated.

.

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