stationary states with a single eigenvalue with positive real part and eigenvector (1,-1,0,...,0)

Question

Suppose we have a system of nonlinear ODEs over the variables x,y,.... which range over the positive reals. Suppose we have a stationary state S whose jacobian matrix has a single eigenvalue with positive real part. And finally suppose that we know that the algebraic multiplicity of the later eigenvalue is equal to 1, and that the associated eigenvector is equal to (1,-1,0,...,0). Can one claim that perturbations of S generate dynamics along which the quantity represented by either x or y vanishes, while the quantity represented by the other variable grows?