Integrable models are special quantum many-body systems because they posses a large number of conserved charges, which enables one to treat these models with analytic tools. However, in usual integrable systems (e.g. XXZ model) certain computations - for example the calculation of correlation functions - are very difficult to tackle exactly. Therefore in recent years ,,simpler'' integrable models were constructed and treated. In my talk I will consider one such system, the so called folded XXZ model. I will present some results regarding the dynamics of this system and its connection to hard rod deformations and other simple integrable systems.