Problem Solving Ideas: Let's first define a complex mathematical formula to know that a, b, c, d, e, f, and g are all integers, and a, b, and c are not all 0, and solve the integer solution of the ternary quadratic indefinite equation ax by cz dxyexzfyz=g (x, y, z). The general way to solve this type of problem is to make use of the congruence theory in mathematics, the properties of modular operations, and the extended Chinese remainder theorem. Since this problem is a ternary quadratic indefinite equation, its solution is more complex and cumbersome than that of ordinary quadratic equations. Step 1: Simplification ProblemTry to simplify the equation by variable substitution or factorization. If d=2ab, e=2ac, f=2bc, you can combine the xy, xz, and yz terms into a perfectly square form, i.e., (abc)(xyz) -2(axbycz)(xyz)(ax by cz -g)=0. However, this is not always successful, and the actual solution needs to be analyzed on a case-by-case basis. Step 2: Special case treatmentIf d=e=f=0, the equation becomes three independent quadratic equations, which can be solved separately. However, when d, e, and f are not all zeros, more complex solutions need to be considered. Step 3: Introduce auxiliary variables, assume new variables t=xy, u=yz, v=zx, and the original equation can be converted into a polynomial system of equations about t, u, v by substitution, and then the system of equations is studied. Step 4: Using the congruence theory and the Chinese remainder theorem, for each binary linear combination (e.g., at btc=0), we can treat it as a quadratic congruence equation with respect to t and solve it at modulo with an appropriate large prime p. Then, according to the Chinese remainder theorem, find the t-value that satisfies the integer solutions of all congruence equations. Step 5: After the possible t value is found in the retrogression verification and enumeration search, it is returned to the original equation, the corresponding y and z values are solved, and then the original equation and integer conditions are satisfied. If the current t-value fails to obtain an integer solution, continue to enumerate other possible t-values until all integer solutions are found. It is important to note that this process can be complex and does not always lead to a specific closed-form solution. For most of these equations, it may be necessary to use computer algorithms to iterate through and filter the solution of integers that meet the criteria. It is also possible that the equation does not have an integer solution, in which case this needs to be proved by counterproof or other mathematical means. 