  We consider a general class of discrete, two-dimensional Markov chains
modeling the dynamics of a population with two types, without mutation or
immigration, and neutral in the sense that type has no influence on each
individual's birth or death parameters. We prove that all the eigenvectors of
the corresponding transition matrix or infinitesimal generator \Pi\ can be
expressed as the product of "universal" polynomials of two variables, depending
on each type's size but not on the specific transitions of the dynamics, and
functions depending only on the total population size. These eigenvectors
appear to be Dirichlet eigenvectors for \Pi\ on the complement of triangular
subdomains, and as a consequence the corresponding eigenvalues are ordered in a
specific way. As an application, we study the quasistationary behavior of
finite, nearly neutral, two-dimensional Markov chains, absorbed in the sense
that 0 is an absorbing state for each component of the process.
