  We consider the irregular (in the Birkhoff and even the Stone sense)
transmission eigenvalue problem of the form $-y''+q(x)y=\rho^2 y,$
$y(0)=y(1)\cos\rho a-y'(1)\rho^{-1}\sin\rho a=0.$ The main focus is on the
''most'' irregular case $a=1,$ which is important for applications. The
uniqueness questions of recovering the potential $q(x)$ from transmission
eigenvalues were studied comprehensively. Here we investigate the solvability
and stability of this inverse problem. For this purpose, we suggest the
so-called regularization approach, under which there should first be chosen
some regular subclass of eigenvalue problems under consideration, which
actually determines the course of the study and even the precise statement of
the inverse problem. For definiteness, by assuming $q(x)$ to be a
complex-valued function in $W_2^1[0,1]$ possessing the zero mean value and
$q(1)\ne0,$ we study properties of transmission eigenvalues and prove local
solvability and stability of recovering $q(x)$ from the spectrum along with the
value $q(1).$ In Appendices, we provide some illustrative examples of regular
and irregular transmission eigenvalue problems, and also obtain necessary and
sufficient conditions in terms of the characteristic function for solvability
of the inverse problem of recovering an arbitrary real-valued square-integrable
potential $q(x)$ from the spectrum, for any fixed $a\in{\mathbb R}.$
