Comment on 'invariants of differential equations defined by vector fields'

In Ndogmo ( 2008 J. Phys. A: Math. Theor. 41 025207), the author claims to have determined a complete list of functionally independent differential invariants up to order 2 for the equivalence group of differential equations defined by vector fields when there are two and three independent variables. In this comment, we show that this is not the case. Using the equivariant moving frame method, we derive a complete set of functionally independent differential invariants of orders 1 and 2 for an arbitrary number n >= 2 of independent variables. In the particular case n = 2, we obtain six functionally independent invariants, two of which were not found in Ndogmo ( 2008 J. Phys. A: Math. Theor. 41 025207). In the case n = 3, we get 21 functionally independent invariants, six of which are new. We also give a complete classification of the differential invariants.