This paper deals with the general Hilbert-Poincaré problem with Carleman’s shift for analytic functions \[ (\partial /\partial \bar z)\Phi (z)=0,\quad z\in S^+,\quad Re\{L\Phi (t_ 0)\}=f(t_ 0),\quad t_ 0\in \Gamma \] where \(S^+\) is a simply connected domain with boundary \(\Gamma\), L is an integral differential operator defined on \(\Gamma\), \[ L\Phi (t_ 0)\equiv \sum^{m}_{j=0}\{a_ j(t_ 0)\Phi^{(j)}(t_ 0)+b_ j(t_ 0)\Phi^{(j)}[\alpha (t_ 0)]+ \]
\[ \int_{\Gamma}h_ j(t_ 0,t)\Phi^{(j)}(t)ds+\int_{\Gamma}k_ j[\alpha (t_ 0),t]\Phi^{(j)}(t)ds\}, \] ds is the differential of arc length, \(\alpha\) (t) is a bijective mapping from \(\Gamma\) onto itself, and \(\alpha\) (t) satisfies \[ \alpha (\alpha (t))=t,\quad \alpha '(t)\neq 0. \] Using the integral representation of analytic functions the above problem is reduced to an equivalent singular integral equation with Carleman’s shift. Then the author obtains the formula of index and finds out necessary and sufficient conditions for the solvability of the original problem. Moreover, the generalized Hilbert-Poincaré problem with two Carleman’s shifts is considered as well.