Summary: In the domain \(D= \{(x, y, z): 0< x,\infty< y<+\infty, -\infty< z<+\infty\}\) it is considered elliptic type equation with singular coefficient \[ u_{xx}+ u_{yy}+ u_{zz}+ {2\alpha\over x} u_x+ \lambda^2 u= 0,\;0< 2\alpha< 1,\;\lambda= \lambda_1+ i\lambda_2, \lambda_1,\lambda_2\in \mathbb{R}. \] Fundamental solutions that express through confluent hypergeometric functions of Kummer \(H_3(a,b; c;x, y)\) from two arguments were found for the given equation. By means of expansion confluent hypergeometric functions of Kummer it is proved, the constructed solutions have a singularity of the order \(1/r\) at \(r\to 0\). Further, in case of when \(\lambda^2= -\mu^2\) for the certain equation by means of found fundamental solutions, boundary value problems are solved in the domain \(D\).