The structure of the analytic solution of the Cauchy problem for a linear partial differential equation of the third order \[ (*)\quad \partial^ 3u/\partial t^ 3-\partial^ 2u/\partial t^ 2-\partial u/\partial t=Du+\sum^{n}_{j=0}a_ j(z)t^ j,\quad \partial^ iu/\partial t^ i|_{t=0}=\phi_ i(z)\quad(i=0,1,2) \] is investigated and the following main result is obtained in the complex field. Fundamental theorem: If (i) \(a_ j(z) (j=0,1,...)\), \(\phi_ i(z)\) and \(b_{m_ 1...m_ N}(z) (0=m_ 1+m_ 2+...+m_ N\leq 3)\) are analytic functions in the neighbourhood of the origin in \(C^ N\); (ii) D is a general third-order differential operator in the variable \(Z=(z_ 1,z_ 2,...,z_ N)\) with holomorphic coefficients; (iii) the series \(\sum^{n}_{j=0}a_ j(z) t^ j\) is uniformly convergent in the neighbourhood of the origin in \(C^{N+1}\) when n is infinite. Then the analytic solution of the problem (*) is given explicitly.