We consider a class of functional equations with one operational symbol which is assumed to be a quasigroup. Equations are quadratic, level and have four variables each. Therefore, they are of the form $x_1x_2\cdot x_3x_4=x_5x_6\cdot x_7x_8$ with $x_i\in\{x,y,u,v\}$ ($1\leq i\leq 8$) with each of the variables occurring exactly twice in the equation. There are 105 such equations. They separate into 19 equivalence classes defining 19 quasigroup varieties. The paper (partially) generalizes the results of some recent papers of Förg-Rob and Krapež, and Polonijo.