  A $n\times n$ matrix $A$, which has a certain sign-symmetric structure
($J$--sign-symmetric), is studied in this paper. It is shown that such a matrix
is similar to a nonnegative matrix. The existence of the second in modulus
positive eigenvalue $\lambda_2$ of a $J$--sign-symmetric matrix $A$, or an odd
number $k$ of simple eigenvalues, which coincide with the $k$-th roots of
$\rho(A)^k$, is proved under the additional condition that its second compound
matrix is also $J$--sign-symmetric. The conditions when a $J$--sign-symmetric
matrix with a $J$--sign-symmetric second compound matrix has complex
eigenvalues, which are equal in modulus to $\rho(A)$, are given.
