On the internal approach to differential equations 3. Infinitesimal symmetries

The geometrical theory of partial differential equations in the absolute sense, without any additional structures, is developed. In particular the symmetries need not preserve the hierarchy of independent and dependent variables. The order of derivatives can be changed and the article is devoted to the higher-order infinitesimal symmetries which provide a simplifying “linear approximation” of general groups of higher-order symmetries. The classical Lie’s approach is appropriately adapted.