The main result of this very saturated note is the following theorem. Let the coefficients \(P_{\ell}(z)\) of the equation \[ (1)\quad P_ 0(z)w^ n+P_ 1(z)w^{n-1}+...+P_ n(z)=0 \] admit in an angle \(S_{\beta}\equiv S_{0,\beta}\), \(S_{\phi,\beta}=\{z:\) \(| \arg z- \phi | \pi)\) and let the set \(\{\lambda_{j\ell}\}\) be separated. Then for some \(\alpha <\beta\) the branches of the solution of (1) in \(S_{\alpha}\) admit each on its own Riemann surface asymptotic expansions \[ w_ k(z)\sim \sum^{r_ k}_{p=0}d_{kp}(z)\exp \{- \mu_ pz\},\quad k=1,...,n,\quad r_ k<\infty, \] where exponents \(\mu_ p\) are represented over \(\lambda_{j\ell}\) and \(d_{kp}\) are meromorphic functions of degree zero. As consequence the authors received the following generalization of the Ritt theorems. If the entire function \(w=w(z)\) satisfies to (1) with \[ P_ j(z)=\sum^{\mu_ j}_{k=0}c_{j\ell}(z)e^{- \lambda_{j\ell}z}, \] \(c_{j\ell}\) are polynomials, then \(w=[Q(z)]^{-1}f(z)\), where Q is a polynomial, and f is a P-quasi- polynomial with special indices.