There is no way of adding equalities and ending up with a surplus. Energy is always conserved. Anyone who thinks otherwise is like the gambler who is looking for some way of combining bets at roulette so as to come out ahead. Every bet pays a percentage to the casino. You can't combine losses and get a gain, no matter how clever you are.
Perpetual motion exists in one sense.
If something has no way of losing energy then it can continue in motion indefinitely. The planets are going to continue to circle the Sun for a long time and atoms and molecules are continuously vibrating. Unfortunately, you can't extract useful work from something which is moving without using up its energy. However, ignorant people are still trying to build machines which generate energy. An early favorite was the overbalanced wheel.