I recently read in Bergeron's book that, for geometric reasons, the hyperbolic Laplacian $$\Delta = -y^2\left(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}\right)$$ commutes with the isometries $g \in \text{SL}_2(\mathbb{R})$ of the upper-half plane, in the sense that $$\Delta L_g = L_g \Delta$$ where $(L_gf)(z) := f(g(z))$.
I tried to show this fact directly, and I found that any holomorphic function $g(z)$ with $v(z) := \Im(g(z))$ satisfying $$v(z)^2 = y^2|\nabla v|^2$$ also commutes with the Laplacian.
This made me wonder, are there more harmonic functions $v(z)$ besides $\Im(g(z)), g \in \text{SL}_2(\mathbb{R})$, satisfying this?
Thanks in advance!