The dynamics of the Ehrhard-M\"uller system with invariant algebraic surfaces

In this paper we study the global dynamics of the Ehrhard-M\"uller differential system $$\dot{x} = s(y - x), \quad \dot{y} = rx - xz - y + c, \quad \dot{z} = xy - z,$$ where $s$, $r$ and $c$ are real parameters, and $x$, $y$, and $z$ are real variables. We classify the invariant algebraic surfaces of degree $2$ of this differential system. After we describe the phase portraits in the Poincar\'e ball of this differential system having one of this invariant algebraic surfaces. The Poincar\'e ball is the closed unit ball in $\mathbb{R}^3$ whose interior has been identified with $\mathbb{R}^3$, and his boundary, the $2$-dimensional sphere $\mathbb{S}^2$, has been identified with the infinity of $\mathbb{R}^3$. Note that in the space $\mathbb{R}^3$ we can go to infinity in as many as directions as points has the sphere $\mathbb{S}^2$. A polynomial differential system as the Ehrhard-M\"uller system can be extended analytically to the Poincar\'e ball, in this way we can study its dynamics in a neigborhood of infinity. Providing these phase portraits in the Poincar\'e ball we are describing the dynamics of all orbits of the Ehrhard-M\"uller system having an invariant algebraic surface of degree $2$.