Common zeros of inward vector fields on surfaces

A vector field X on a manifold M with possibly nonempty boundary is inward if it generates a unique local semiflow ${\mathrm{\Phi }}^X$. 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 $C^1$ vector fields on surface M such that $[X,Y]\wedge X=0$ 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 $C^2$ and ${\mathrm{\Phi }}^Y$ preserves area. Applications are made to actions of Lie algebras and groups.