While there are many scientific manuscripts devoted to various details of the evolution of curves and their models, there are still many perspectives of the research subject that required comprehensive investigation. In particular, this manuscript is devoted to the search of a new class of time evolution equations of space curves written in the anholonomic coordinates. In this way, we have direct observation and access to form a new class of anholonomic surfaces induced by the given evolution systems. Thus, the proper representation of the evolution equations allows us to obtain useful characteristics of the corresponding evolution dynamics.