From the text: The present paper continues the series [the author, Differ. Uravn. 35, No. 8, 1094-1100 (1999; Zbl 0973.35085); M. E. Lerner and O. A. Repin, ibid. 35, No. 8, 1087-1093 (1999; Zbl 0976.35028); ibid. 37, No. 11, 1562-1564 (2001; Zbl 1016.35016)]. For the degenerate elliptic equation \[ y^m u_{xx}+ u_{yy}- b^2 y^m u= 0,\qquad 0< x< 1,\quad y> 0,\quad b>0, \] with the nonlocal condition \[ u(0,y)= u(1,y),\qquad u_x(0,y)= 0,\quad y> 0, \] and the condition \[ u(x,0)= f(x),\quad f\in C^{2+z}[0,1],\quad f(0)= f(1),\quad f'(0)= 0, \] it is shown that there exists a unique solution \[ u\in C^0([0,1]\times [0,\infty])\cap C^2((0,1)\times (0,\infty) \] tending to zero at infinity. Moreover, a series representation of this solution is given as follows: \[ u(x,y)= u_0(y)+ \sum^\infty_{n=1} u_n(y)\cos 2\pi nx+ \sum^\infty_{n=1} v_n(y)x\sin 2\pi nx, \] where for example \[ v_n(y)= {8\over\Gamma(1/(2q))} \Biggl({\sqrt{(2\pi u)^2+ b^2}\over 2q}\Biggr)^{1/(2q)} \int^1_0 f(t)\sin 2\pi nt dt\sqrt{y} K_{1/(2q)}\Biggl({\sqrt{2\pi n)^2+ b^2}y^q\over q}\Biggr), \] here \(K_{1/(2q)}\) is a so-called Macdonald function.