Summary: The Fleishman system of distributions is often used in simulation experiments to generate non-normal distributions with given skewness \(\gamma_1\) and kurtosis \(\gamma_2\). A Fleishman distribution is generated from a standard normal random variable \(\boldsymbol{x}\) (expectation \(\mu = 0\) and variance \(\sigma^2 = 1)\) by the cubic random function \(\boldsymbol{y} = a + b\boldsymbol{x} + c\boldsymbol{x}^2 + d\boldsymbol{x}^3\) with real parameters \(a, b, c\) and \(d\). For appropriate pairs \((\gamma_1, \gamma_2)\) members of the system exist. The coefficients can be calculated from a nonlinear system of algebraic equations. We are interested in the different real solutions. Solvability conditions and properties of solutions are discussed. The mathematical software MATLAB is used to compute special solutions and to get some insight into the solution structure.