  A new class of sign-symmetric matrices is introduced in this paper. Such
matrices are named J--sign-symmetric. The spectrum of a J--sign-symmetric
irreducible matrix is studied under assumptions that its second compound matrix
is also J--sign-symmetric and irreducible. The conditions, when such matrices
have complex eigenvalues on the largest spectral circle, are given. The
existence of two positive simple eigenvalues $\lambda_1 > \lambda_2 > 0$ of a
J--sign-symmetric irreducible matrix A is proved under some additional
conditions. The question, when the approximation of a J--sign-symmetric matrix
with a J--sign-symmetric second compound matrix by strictly J--sign-symmetric
matrices with strictly J--sign-symmetric compound matrices is possible, is also
studied in this paper.
