Question concerning phase curves and Lissajous figures

I want to draw the "orbits" of a spherical pendulum under small oscillations. In this case its equations are given by $ddot{x}_{1}=-x_{1}$ and $ddot{x}_{2}=-x_{2}$. Of course the potential energy is given by $U=frac{1}{2}(x_{1}^2+x_{2}^2)$, and the level sets of it will be concentric circles in the $x_{1}x_{2}$ plane. Consequently, by the law of conservation of energy:
$$E=frac{1}{2}(dot{x}_{1}^2+dot{x}_{2}^2)+frac{1}{2}(x_{1}^2+x_{2}^2)={rm const}.$$
And this represents a sphere in four space. Now, suppose I want to draw the orbits in the $x_{1}$–$x_{2}$ plane. Is all I have to do to make $dot{x}_{1}=dot{x}_{2}=0$? If so, I also get concentric spheres for each value of E.

But now using the Lissajous figures method: the solutions of these equations can be written as $x_{1}=A_{1}sin{(t+phi_{1})}$ and $x_{2}=A_{2}sin{(t+phi_{2})}$, where $A_{i}=sqrt{2E}$ for $i=1,2$. So the orbits have to lie inside this region. By Lissajous figures, the shape of the curve on the $x_{1}$–$x_{2}$ plane depends entirely on the difference $phi_{2}-phi_{1}$ and this is a circle in the case where $phi_{2}-phi_{1}=pi/2$.

So, what shape exactly does this orbit have? I may be misunderstanding some concepts, and if so, please let me know. Thanks in advance.