A vector field X on a manifold M with possibly nonempty boundary is inward if
it generates a unique local semiflow ΦX. A compact relatively open set K
in the zero set of X is a block. The Poincar\'e-Hopf index is generalized to an
index for blocks that may meet the boundary. A block with nonzero index is
essential.
Let X, Y be inward C1 vector fields on surface M such that [X,Y]∧X=0 and let K be an essential block of zeros for X. Among the main results are
that Y has a zero in K if X and Y are analytic, or Y is C2 and ΦY
preserves area. Applications are made to actions of Lie algebras and groups