The Cauchy problem for the generalized hyperbolic Novikov-Veselov equation via the Moutard symmetries

We begin by introducing a new procedure for construction of the exact solutions to Cauchy problem of the real-valued (hyperbolic) Novikov-Veselov equation which is based on the Moutard symmetry. The procedure shown therein utilizes the well-known Airy function $\text{Ai}(\xi )$ which in turn serves as a solution to the ordinary differential equation $\frac{d^{2}z}{d{\xi }^2}=\xi z$. In the second part of the article we show that the aforementioned procedure can also work for the $n$-th order generalizations of the Novikov-Veselov equation, provided that one replaces the Airy function with the appropriate solution of the ordinary differential equation $\frac{d^{n-1}z}{d{\xi }^{n-1}}=\xi z$.