Do classical spin systems with the same metastable states have identical hamiltonians?

The question whether systems having the same set of metastable states have identical Hamiltonian and hence identical dynamics is addressed and examined for various classical spin systems. The answer depends on the number of metastable states and on the distribution of the local fields for small fields. The rule that static properties determine dynamical properties is found to be applicable for a large class of random systems, almost any random system in the mean-field limit and many random systems in finite number of dimensions.