Liquid hydrazine, N2H4, has been used as a fuel for rockets. When the rocket is to be launched, a catalyst causes the liquid hydrazine to decompose quickly into elemental nitrogen and hydrogen gases. The rapid expansion of the product gases and the heat released by the reaction for the product gases and the heat released by the reaction provide the thrust for the rocket. Balanced equation.
Interesting but I don't see a question.
If it just splits in the presence of a catalyst, I'd guess
N2H4 = N2 + 2H2
But it burns with oxygen in the reaction
N2H4 + O2 = N2 + 2H2O