On the independence of equations in three variables

We prove that an independent system of equations in three variables with a nonperiodic solution and at least two equations consists of balanced equations only. For that, we show that the intersection of two different entire systems contains only balanced equations, where an entire system is the set of all equations solved by a given morphism. Furthermore, we establish that two equations which have a common nonperiodic solution have the same set of periodic solutions or are not independent.