  The general solution of the intertwining relations between a pair of
Schr\"odinger Hamiltonians by the supercharges of third order in derivatives is
obtained. The solution is expressed in terms of one arbitrary function. Some
properties of the spectrum of the Hamiltonian are derived, and wave functions
for three energy levels are constructed. This construction can be interpreted
as addition of three new levels to the spectrum of partner potential: a ground
state and a pair of levels between successive excited states. Possible types of
factorization of the third order supercharges are analysed, the connection with
earlier known results is discussed.
