As you say in your first line, the eigenvalues are 1 and -8 so there is no eigenvector for . Was that a typo?
By definition, if is an eigenvector corresponding to eigenvalue -8, then we must have
That is equivalent to the two equations -5x+ 3y= -8x and 6x- 2y= -8y which then give 3y= -3x and 6x= -6y both of which are equivalent to y= -x. Any vector of the form , which, of course, includes , is an eigenvector corresponding to eigenvalue -8.