corrections: As the evil Mike Speciner pointed out, f(z)=e is a counterexample. The correction: Theorem should specify NONCONSTANT analytic functions. Another error was I said the terminating-decimal complex numbers z with |z|, Re z, Im z, all terminating, was a "subring" I should have said "subsubgroup." Re Henry Baker's desires: they are difficult. First, you want a subFIELD while what I said only gave a multiplicative subGROUP (cannot add). Second, even if you had that, you'd still have problems since the sizes of the terminating decimals (or whatever) would grow huge. Third, there already is a notion in the DiffEq world, "symplectic integrators" which are intended to preserve certain constants of motion when simulating Hamiltonian systems, albeit I think in a weaker sense that HB had in mind.