  In this paper we define $q$-spherical surfaces as the surfaces that contain
the absolute conic of the Euclidean space as a $q-$fold curve. Particular
attention is paid to the surfaces with singular points of the highest order.
Two classes of such surfaces, with one and two $n-$fold points, are discussed
in detail. We study their properties, give their algebraic equations and
visualize them with the program {\it Mathematica}.
