Symmetries of Schrödinger-Pauli equations for charged particles and
quasirelativistic Schrödinger equations  111E-mail<EMAIL_ADDRESS>
A. G. Nikitin Institute of Mathematics, National Academy of Sciences of
Ukraine,
3 Tereshchenkivs’ka Street, Kyiv-4, Ukraine, 01601

Lie symmetries of Schrödinger-Pauli equations for charged particles and
quasirelativistic Schrödinger equations are classified. In particular a new
superintegrable system with spin-orbit coupling is discovered.

##  1 Introduction

Staring with the appearance of the classical works of Weil, Wigner and Wan der
Warden everybody understands the dominant role played by symmetries in quantum
mechanics. In particular all fundamental quantum mechanical systems such as
Hydrogen atom and harmonic oscillator can be specified as ones having the most
extended number of integrals of motion which can be admitted by Hamiltonian
systems, and this property is identified as the maximal superintegrability.

Historically the systematic search for superintegrable systems started with
the papers of Winternitz with collaborators [1], [2] and [3] where the
complete description of 2d quantum mechanical systems admitting second order
integrals of motion was presented. And it needs as much as twenty four years
to extend this result to the case of the 3d systems, see papers [4], [5] and
[6]. Notice that a particular subclass of the 2d systems invariant with
respect to rotations with respect to the third coordinate axis was considered
already in [3].

The second (and higher) order symmetries can be related to such nice
properties of Schrödinger equation (SE) as superintegrability and
supersymmetry, see surveys [7] and [8]. We will not discuss them here but
mention that searching for such symmetries is still a very popular business,
and the modern trends in this field are related to the third order and even
arbitrary order integrals of motion [9, 10, 11], see also paper [12] where the
determining equations for such symmetries were presented.

It was Winternitz again [13, 14, 15] who starts and continued the study of
superintegrable Schrödinger equations (SEs) generalized by the presence of the
spin-orbit coupling term. The principally new feature of such equations is
that they include matrix potentials with generate essential complications of
the related classification procedure.

The spin-orbit coupling is a well grounded and experimentally confirmed effect
which can be treated as a relativistic correction of order $\frac{1}{c^{2}}$
where $c$ is the speed of light. But there exist one more correction which is
more essential since its order is $\frac{1}{c}$. This correction is described
by the Pauli term which include the scalar product of the spin and magnetic
fields vectors. Thus it was natural to classify superintegrable Schrödinger-
Pauli (SP) equations. We can mention papers devoted to its supersymmetries
[16, 17], supersymmetries and higher order symmetries [18, 19, 20] and Fock
symmetries [21, 22, 23].

Rather paradoxically, the less complicated (but not less important!) Lie
symmetries of the mentioned equations are still terra incognita. More exactly,
everything is complete with the symmetries of the standard SE whose Lie
symmetries were classified long time ago in papers [24], [25] and [26].
Symmetries of the 3d SE with non-trivial scalar potential were described by
Boyer [26], whose results where corrected in [27]. Symmetries of this equation
with scalar and vector potentials have been classified in [28]. Symmetries of
SE with a scalar (but matrix) potential are described in [29]. Let us mention
also that the complete group classification of SE equation with position
dependent mass appears rather recently in papers [30] and [31, 32] for the
stationary and time dependent equations correspondingly. For the
superintegrability aspects of such equations see papers [33] and [34], a
superintegrable relativistic system is discussed in [35].

In the present paper we give the completed description of all inequivalent
continuous symmetries which can be accepted by time dependent SP equation for
charged particles. In addition, we classify also Lie symmetries of the quasi
relativistic Schrödinger equation which includes both Pauli, spin-orbit and
Darwin interaction terms. In this way we present a certain group-theoretical
grounds for all quantum mechanical systems whose higher symmetries have been
classified in the above cited papers. In particular, we give a priori
information concerning symmetries which can be accepted by the related quasi
relativistic models. Secondly, the related symmetry groups supply us by the
equivalence relations between the systems whose higher order integrals of
motion are classified [36]. Finally, just Lie symmetries present powerful
tools for construction of exact solution which sometimes have a very high
value.

##  2 Schrödinger-Pauli equation for charged particles

Let us consider the standard SP equation for a charged particle with spin 1/2
interacting with the external electromagnetic field and write it in the
following form:

$\displaystyle\left({\mathrm{i}}\frac{\partial}{\partial
t}-H\right)\psi(t,{\bf x})=0$ (1)

The corresponding Hamiltonian $H$ looks as follows:

$\displaystyle H=\frac{1}{2}\pi_{a}\pi_{a}+A^{0}+g\sigma_{a}H^{a}$ (2)

where $\pi_{a}=p_{a}-eA^{a},\
p_{a}=-{\mathrm{i}}\frac{\partial}{\partial{x_{a}}}$, $A^{0}$ and $A^{a}$ are
components of the vector-potential of the electromagnetic field, $e$ and $g$
are coupling constants,

$\displaystyle H^{a}=\varepsilon_{abc}\frac{\partial A^{c}}{\partial x_{b}}$
(3)

are components of the external magnetic field strengths, $\sigma_{a}$ are
Pauli matrices and summation is imposed over the repeating indices $a$ over
the values $a=1,2,3$. Moreover, the vector-potential of the external field is
supposed to be time independent.

The vector-potential is supposed to satisfy the continuity equation which, in
view of its time independence, is reduced to the divergenceless condition for
the vector $A^{a}$. However in paper [28] we prefer to change it by the
condition $A^{3}=0$ which always can be done using the gauge transformations.
In particular we can set:

$\displaystyle A^{3}=0,\quad A^{1}=\partial_{1}\tilde{F}({\bf
x})+\partial_{2}G({\bf x}),\quad A^{2}=\partial_{2}F({\bf
x})-\partial_{1}G({\bf x})$ (4)

where $\partial_{1}=\frac{\partial}{\partial x_{1}}$, etc, $\tilde{F}$ and $G$
are functions of $\bf x$.

Alternatively using the gauge transformation
$H\to\exp(-{\mathrm{i}}\tilde{F}({\bf x}))H\exp({\mathrm{i}}\tilde{F}({\bf
x}))$ we can transform (4) to a more compact form

$\displaystyle A^{1}=\partial_{2}G({\bf x}),\quad A^{2}=-\partial_{1}G({\bf
x}),\quad A^{3}=F(\bf x)$ (5)

where $F({\bf x})=\partial_{3}\tilde{F}({\bf x})$. Just representation (5)
will be used in the present paper.

Substituting (5) into (2) we reduce it to the following form:

$\displaystyle
H=\frac{1}{2}p_{a}p_{a}-\frac{e}{2}(A^{a}\partial_{a}+\partial_{a}A^{a})+V$
(6)

where $V$ is the effective matrix potential:

$\displaystyle\begin{split}&V=V^{0}+g\sigma_{a}H^{a},\\\
&V^{0}=A^{0}+\frac{e^{2}}{2}((A^{1})^{2}+(A^{2})^{2}+(A^{3})^{2})\end{split}$
(7)

and

$\displaystyle H^{1}=\partial_{2}F+\partial_{1}\partial_{3}G,\quad
H^{2}=\partial_{2}\partial_{3}G-\partial_{1}F,\quad
H^{3}=-(\partial_{1}^{2}+\partial_{2}^{2})G.$ (8)

Equation (1) with Hamiltonian (6) includes three arbitrary elements, i.e.,
functions $A^{0}$, $F$ and $G$.

###  2.1 Determining equations

Let us search for symmetries of the SP equation with respect to continuous
groups of transformations of dependent and independent variables. Bearing in
mind the linearity of equation (1), (6) we can represent the generator of the
symmetry group in the form

$\displaystyle
Q=\xi^{0}\partial_{t}+\xi^{a}\partial_{a}+\tilde{\eta}\equiv\xi^{0}\partial_{t}+\frac{1}{2}\left(\xi^{a}\partial_{a}+\partial_{a}\xi^{a}\right)+{\mathrm{i}}\eta,$
(9)

where $\tilde{\eta}=\frac{1}{2}\xi^{a}_{a}+{\mathrm{i}}\eta,\ \ $ $\xi^{0}$,
$\xi^{a}$ and $\eta$ are functions of $t,{\mathbf{x}}$ and
$\partial_{t}=\frac{\partial}{\partial t}$. Moreover, $\eta$ is a $2\times 2$
matrix which can be expanded via Pauli matrices:

$\displaystyle\eta=\eta^{0}\sigma_{0}+\sigma_{a}\eta^{a}$ (10)

where $\sigma_{0}$ is the $2\times 2$ unit matrix which will be omitted in the
following formulae. In contrary, $\xi^{0}$ and $\xi^{a}$ are scalar functions
which are multipliers for the unit matrix.

Generator (9) transforms solutions of equation (1), (6) into solutions
provided the following operator equation is satisfied:

$\displaystyle[Q,L]\equiv QL-LQ=\alpha L$ (11)

where $L={\mathrm{i}}\partial_{t}-H$ and $\alpha$ is an unknown function of
$t$ and $\bf x$.

Evaluating the commutator in the l.h.s. of (11) and equating coefficients for
the linearly independent differentials we obtain the following system of
equations for unknowns $\xi^{0},\xi^{a},\eta,\ V$ and $\alpha$:

$\displaystyle\dot{\xi}^{0}=-\alpha,\quad\xi^{0}_{a}=0,$ (12)
$\displaystyle\xi^{b}_{a}+\xi^{a}_{b}-\frac{2}{3}\delta_{ab}\xi^{i}_{i}=0,$
(13) $\displaystyle 2\xi^{i}_{i}=-3\alpha,$ (14)
$\displaystyle\dot{\xi}^{a}+\eta_{a}=e(A^{b}\xi^{a}_{b}-\xi^{b}A^{a}_{b}+\alpha
A^{a}),$ (15) $\displaystyle\xi^{a}V_{a}=\alpha
V+\dot{\eta}+{\mathrm{i}}[\eta,V]-eA^{a}\eta_{a}$ (16)

where $\dot{\eta}=\frac{\partial\eta}{\partial t}$ and
$\eta_{a}=\frac{\partial\eta}{\partial x_{a}}$.

Thus to find symmetries of SP equation we are supposed to solve the system of
equations (12)–(16) which is rather complicated. In particular this system
includes two matrix equations (15) and (16). Collecting in (15) the terms
proportional to the unit matrix we obtain:

$\displaystyle\dot{\xi}^{a}+\eta^{0}_{a}=e(A^{b}\xi^{a}_{b}-\xi^{b}A^{a}_{b}+\alpha
A^{a})$ (17)

while the remaining terms are reduced to the following condition:

$\displaystyle\frac{\partial{\eta^{a}}}{\partial x_{b}}=0.$ (18)

Equation (16) also is decoupled to the scalar and vector parts:

$\displaystyle\xi^{a}V^{0}_{a}=\alpha
V^{0}+\dot{\eta}^{0}-eA^{a}\eta^{0}_{a},$ (19)

and

$\displaystyle g(\xi^{a}H_{a}^{b}-\alpha
H^{b}+2\varepsilon^{bcd}\eta^{c}H^{d})-\dot{\eta}^{b}=0$ (20)

where $\varepsilon^{bcd}$ is the absolutely antisymmetric unit tensor.

Substituting (17) into (19) and using definition (7) we reduce the latter
equation to the following form:

$\displaystyle\xi^{a}A^{0}_{a}+\dot{\xi}^{a}A^{a}=\alpha
A^{0}+\dot{\eta}^{0}.$ (21)

In other words, system (12)–(16) includes the autonomous subsystem formed by
equations (12), (13), (14), (17) and (19). Its solutions, which give
symmetries of equation (1), (6) in the particular case $g=0$, have been
already obtained in paper [28]. On the other hand setting
$A^{1}=A^{2}=A^{3}=0$ we reduce our classification problem to description of
symmetries of the SP equation for chargeless particles. This description had
been made in paper [29]. The more general problem with nontrivial coupling
constant $g$ and nontrivial vector potentials is solved in the present paper.

###  2.2 Classification of symmetries

In accordance with the above the first necessary step in description of
symmetries of the SP equation for charged particles is the group
classification of the standard SEs including the vector potential of the
external electromagnetic field. Then going over all inequivalent versions of
such external fields found in paper [28] we will select such of them which are
compatible with equation (20) . By construction equations (1), (2) with
arbitrary potentials admit two dimensional symmetry algebra spanned on
$P_{0}={\mathrm{i}}\frac{\partial}{\partial t}$ and the unit operator. In
accordance with the results presented in [28] there exist six inequivalent
cases when this algebra is extended to three dimensional one. The
corresponding potentials and the related additional symmetries $Q$ are given
by the following formulae:

$\displaystyle\begin{split}&A^{1}=\partial_{2}G({x_{1},x_{2}}),\quad
A^{2}=-\partial_{1}G({x_{1},x_{2}}),\\\ &A^{3}=F(x_{1},x_{2}),\quad
A^{0}=R(x_{1},x_{2}),\end{split}$ (22) $\displaystyle
P_{3}=p_{3}=-{\mathrm{i}}\partial_{3}$ (23)

where $F(.,.),\ G(.,.)$ and $R(.,.)$ are arbitrary functions of the arguments
fixed in the brackets;

$\displaystyle\begin{array}[]{l}A^{1}=\partial_{2}G(\tilde{r},x_{3})\quad
A^{2}=-\partial_{1}G(\tilde{r},x_{3}),\\\ A^{3}=F(\tilde{r},x_{3}),\quad
A^{0}=R(\tilde{r},x_{3})+\kappa\varphi,\end{array}$ (26) $\displaystyle
Q_{1}=L_{3}+\kappa t$ (27)

where $\tilde{r}=\sqrt{x_{1}^{2}+x_{2}^{2}},\
\varphi=\arctan\frac{x_{2}}{x_{1}},L_{3}=x_{1}p_{2}-x_{2}p_{1}$ and $\kappa$
is an arbitrary real parameter;

$\displaystyle\begin{array}[]{l}A^{1}=\partial_{2}G(\tilde{r},\varkappa),\quad
A^{2}=-\partial_{1}G(\tilde{r},\varkappa),\\\
A^{3}=F(\tilde{r},\varkappa),\quad
A^{0}=R(\tilde{r},\varkappa)+\kappa\varphi,\end{array}$ (30) $\displaystyle
Q_{2}=L_{3}+p_{3}+\kappa t$ (31)

where $\varkappa=\varphi-x_{3}$;

$\displaystyle\begin{array}[]{l}A^{1}=\partial_{2}G(\theta,\mu),\quad
A^{2}=-\ln(r)\partial_{1}G(\theta,{\mu})\\\
A^{3}=\ln(r)(F(\theta,{\mu})+\tilde{F}(\nu,\mu)),\quad
A^{0}=\frac{1}{r^{2}}R(\theta,{\mu}),\end{array}$ (34) $\displaystyle
Q_{3}=D+\alpha L_{3},$ (35)

where $\mu=\alpha\rho{-\varphi},\ \nu=\rho+\alpha\varphi,\
\rho=\ln(\tilde{r})$ and $D=2tP_{0}-x_{a}p_{a}+{3{\mathrm{i}}}/2;$

$\displaystyle\begin{array}[]{l}A^{1}=\partial_{2}G({x_{1},x_{2}}),\quad
A^{2}=-\partial_{1}G({x_{1},x_{2}}),\quad A^{3}=F({x_{1},x_{2}}),\\\
A^{0}=R({x_{1},x_{2}})-\frac{\omega^{2}x_{3}^{2}}{2}+\omega
x_{3}F(x_{1},x_{2}),\end{array}$ (38) $\displaystyle B^{3}=\exp(\omega
t)(p_{3}-\omega x_{3});$ (39)
$\displaystyle\begin{array}[]{l}A^{1}=\partial_{2}G(\theta,\varphi),\quad
A^{2}=-\partial_{1}G(\theta,\varphi),\quad
A^{3}=\frac{1}{r}(\tilde{F}(\theta,\varphi)+(\partial_{\rho}+2)F(\rho,\varphi)),\\\
A^{0}=-\frac{\omega^{2}r^{2}}{2}+\frac{1}{r^{2}}R(\theta,\varphi)+4\omega\partial_{\rho}F(\rho,\varphi),\end{array}$
(42) $\displaystyle A^{+}=\exp(2\omega
t)(p_{0}+\omega^{2}r^{2}-\frac{\omega}{2}(x_{a}p_{a}+p_{a}x_{a})).$ (43)

Notice that $L_{3}$ is nothing but the third component of the orbital moment.
Up to rotation it can be replaced by the first component
$L_{1}=x_{2}p_{3}-x_{3}p_{2}$ or the second component
$L_{2}=x_{3}p_{1}-x_{1}p_{3}$.

Consider consequently all potentials and symmetries (22)–(43) in more detail
and verify their compatibility with equation (20). The vector of magnetic
field corresponding to potentials (22) has the following components (refer to
(8)):

$\displaystyle H^{1}=-\partial_{2}F({x_{1},x_{2}}),\quad
H^{2}=\partial_{1}F({x_{1},x_{2}}),\quad
H^{3}=-(\partial_{1}^{2}+\partial_{2}^{2})G({x_{1},x_{2}}).$ (44)

The related symmetry (23) includes only a part of matrix $\eta$ which is
proportional to the unit matrix, i.e., only the first term from the r.h.s. of
equation (10). Whenever the Pauli term is present, it is necessary to add the
remaining part of $\eta$ equal to $\sigma^{a}\eta^{a}$, i.e., replace (23) by
the following operator:

$\displaystyle\hat{P}_{3}=p_{3}+\eta^{0}+\eta^{a}\sigma^{a}$ (45)

where $\eta^{0}$ and $\eta^{a}$ are unknown functions. The corresponding
condition (20) is reduced to the following form:

$\displaystyle
g(\partial_{3}H^{b}+2\varepsilon^{bcd}\eta^{c}H^{d})-\dot{\eta}^{b}=0.$ (46)

All components of the magnetic field presented in (44) do not depend on
$x_{3}$, the same is true for functions $\eta^{a}$ in view of equation (18).
This means that condition (46) nullifies all functions $\eta^{a},a=1,2,3$ if
$F({x_{1},x_{2}})$ and $G({x_{1},x_{2}})$ are arbitrary. However, for the very
special case of these functions corresponding to constant magnetic field (44)
equation (46) has non-trivial solutions which, up to rotation transformations,
can be represented in the following form:

$\displaystyle\begin{split}&H^{1}=H^{2}=0,\ H^{3}=c_{1},\\\
&\eta^{1}=c_{2}\sin(2c_{1}gt)+c_{3}\cos(2c_{1}gt),\\\
&\eta^{2}=c_{2}\cos(2c_{1}gt)-c_{3}\sin(2c_{1}gt),\eta^{3}=c_{4}\end{split}$
(47)

where $c_{1},...,c_{4}$ are arbitrary parameters. Moreover, up to scaling of
the independent variables and the routine trigonometric manipulations we can
reduce these parameters to the form $c_{1}=1,c_{2}=0,\ c_{3}=1,c_{4}=c$.

The next system we consider is given by relations (26). The corresponding
symmetry (27) again has to be generalised by adding matrix $\eta$ and written
in the form:

$\displaystyle Q=L_{3}+\eta^{0}+\sigma^{a}\eta^{a}.$ (48)

The components $\eta^{a}$ should satisfy condition (20) which is reduced to
the following form:

$\displaystyle
g((x_{1}\partial_{2}-x_{2}\partial_{1})H^{b}+2\varepsilon^{bcd}\eta^{c}H^{d})-\dot{\eta}^{b}=0$
(49)

or

$\displaystyle
g(\partial_{\varphi}H^{b}+2\varepsilon^{bcd}\eta^{c}H^{d})-\dot{\eta}^{b}=0.$
(50)

But in accordance with (8) and (26) the related components of the magnetic
field can be represented as:

$\displaystyle H^{1}=-\partial_{3}(x_{2}G^{1}-x_{1}G^{2}),\quad
H^{2}=\partial_{3}(x_{1}G^{1}+x_{2}G^{2}),\quad
H^{3}=(2+\tilde{r}\partial_{\tilde{r}})G^{2}$ (51)

where $G^{1}$ and $G^{2}$ are functions of $\tilde{r}$ and $x_{3}$. In view of
(51) equations (50) are reduced to the following form:

$\displaystyle 2g(\eta^{1}H^{2}-\eta^{2}H^{1})=\dot{\eta}^{3},$ (52)
$\displaystyle\begin{split}&g(2\eta^{2}H^{3}+(2\eta_{3}-1)H^{1})=\dot{\eta}^{1},\\\
&g(2\eta^{1}H^{3}+(1-2\eta_{3})H^{2})=-\dot{\eta}^{2}.\end{split}$ (53)

For generic magnetic field whose components are given by equation (51) we
obtain from (52) that

$\displaystyle\eta^{1}=0,\quad\eta^{2}=0,\quad\eta^{3}=\frac{1}{2}.$ (54)

However, for the constant magnetic field there is one more solution given by
equations (47).

In the complete analogy with the above it is possible to consider the
remaining systems (30)–(42) and prove, that all symmetries (31), (35), (39)
and (43) valid for the Schrödinger equation are kept for the Schrödinger-Pauli
equation also, provided $L_{3}$ is changed to
$J_{3}=L_{3}+\frac{1}{2}\sigma_{3}$. The main point of the proof is that
$H^{3}$ always depends on invariant variables for these symmetries, and so in
all cases we have the same equation (53) which nullifies either
$\eta^{1},\eta^{2}$ or $H^{1},H^{2}$.

The obtained results can be formulated as the following statement.

Theorem. All Lie symmetries of the Schrödinger equation with time independent
scalar and vector potentials can be extended to the symmetries of the
corresponding SP equation, if we change the orbital momentum operators $L_{a}$
by the total orbital momentum operators $J_{a}=L_{a}+\frac{1}{2}\sigma_{a}$ .
In addition, for constant magnetic fields or for the fields with the only
nontrivial component $H^{3}$, the additional matrix symmetries

$\displaystyle Q=\sigma_{1}\cos(2gt)+\sigma_{2}\sin(2gt)+c\sigma_{3}\quad{\it
and}\quad\tilde{Q}=\sigma_{3}$ (55)

are valid.

Since the symmetries of the Schrödinger equation with time independent scalar
and vector potentials have been described in paper [28], in fact we have in
hands all inequivalent symmetries of the related SP equation. In the following
section we represent them together with the corresponding arbitrary elements
$F,G$ and $A^{0}$.

###  2.3 Classification results

Using Theorem 1 and the results presented in paper [28] we can effectively
describe all inequivalent Lie symmetries of the SP equation. Let us present
them in the following tables. In accordance with [28] the dependence of such
symmetries on time can be linear, quadratic or exponential. In additional,
these symmetries can be time independent.

In the tables $J_{a}=L_{a}+\frac{1}{2}\sigma_{a},a=1,2,3,$ and

$\displaystyle\begin{split}&L_{a}=\varepsilon_{abc}x_{b}p_{c},\\\
&P_{a}=p_{a},\ G_{a}=tP_{a}-x_{a}\end{split}$ (56)
$\displaystyle\begin{split}&D=2tP_{0}-x_{a}p_{a}+{3{\mathrm{i}}}/2,\\\
&A=tD-t^{2}P_{0}+{r^{2}}/2\end{split}$ (57)

are symmetries of the free SE. In addition,

$\displaystyle\begin{split}&A^{+}(\omega)=\exp(2\omega
t)(P_{0}+\omega^{2}r^{2}-\frac{\omega}{2}(x_{a}P_{a}+P_{a}x_{a})),\\\
&B^{\pm}_{a}(\omega)=\exp(\pm\omega t)(P_{a}\mp\omega
x_{a}),a=1,2,3\end{split}$ (58)

are symmetries of SE with repulsive potential. In addition, $F(.),G(.),R(.)$
and $\tilde{R}(.)$ are arbitrary functions of the arguments fixed in the
brackets, and the Greek letters denote arbitrary coefficients.

In contrast with [28], in the classification tables we will specify
inequivalent arbitrary elements $F,G$ and $A^{0}$ instead of direct
presentation of vector potentials. In this way we obtain the tables in a
rather compact form. In addition, we arrange the classification results in the
manner convenient for presentation of symmetries of more general equation
including spin-orbit and Darwin terms. Just these equations are discussed in
the following section.

Table 1. Generating functions and symmetries linear in $t$

No | Functions $F$ | Functions $G$ | Potential $A^{0}$ | Symmetries | Potential $S$
---|---|---|---|---|---
1. | $F(x_{1},x_{2})$ | $G(x_{1},x_{2})$ | $R_{0}(x_{1},x_{2})$ | $P_{3}$ | $\tilde{R}_{0}(x_{1},x_{2})$
2. | $F(\tilde{r},x_{3})$ | $G(\tilde{r},x_{3})$ | $R(\tilde{r},x_{3})+\kappa\varphi$ | $J_{3}+\kappa t$ | $\tilde{R}(\tilde{r},x_{3})$
3. | $F(\tilde{r},\varphi-x_{3})$ | $G(\tilde{r},\varphi-x_{3})$ | $\begin{array}[]{l}R(\tilde{r},\varphi-x_{3})\\\ +\kappa\varphi\end{array}$ | $J_{3}+\kappa t+P_{3}$ | $\tilde{R}(\tilde{r},\varphi-x_{3})$
4. | $\tilde{F}(\tilde{r})$ | $G(\tilde{r})$ | $R(\tilde{r})+\kappa\varphi$ | $P_{3},\ J_{3}+\kappa t$ | $R(\tilde{r})$
5. | $0$ | $G(\tilde{r})$ | $R(\tilde{r})+\kappa\varphi$ | $P_{3},\ J_{3}+\kappa t,\ G_{3}$ | $\tilde{R}(\tilde{r})$
6. | $0$ | $G(x_{1},x_{2})$ | $R(x_{1},x_{2})$ | $P_{3},\ G_{3}$ | $\tilde{R}(x_{1},x_{2})$
7. | $x_{1}F(x_{3})$ | $x_{1}G(x_{3})$ | $R(x_{3})$ | $P_{1},\ P_{2}$ | $\tilde{R}(x_{3})$
8. | $0$ | $G(\tilde{r})$ | $R(\tilde{r})$ | $J_{3},\ P_{3},G_{3}$ | $\tilde{R}(\tilde{r})$
9. | $0$ | $x_{2}\Phi(x_{3})$ | $R(x_{3})$ | $P_{1},\ P_{2},\ G_{2}$ | $\tilde{R}(x_{3})$
10. | $0$ | $x_{3}\varphi/r$ | $R(r)$ | $\begin{array}[]{l}J_{1}+\lambda(x_{3}^{2}\partial_{2}\varphi-x_{1})/r,\\\ J_{2}-\lambda(x_{3}^{2}\partial_{1}\varphi-x_{2})/r,\\\ J_{3}\end{array}$ | $\tilde{R}(r)$
11. | $\begin{array}[]{l}x_{1}F(x_{3})+\alpha x_{1}x_{2}\end{array}$ | $\begin{array}[]{l}x_{1}G(x_{3})+\alpha x_{1}^{2}\end{array}$ | $R(x_{3})$ | $\begin{array}[]{l}P_{1}+\alpha x_{2},P_{2}-\alpha x_{1}\end{array}$ | $\tilde{R}(x_{3})$
12. | $\begin{array}[]{l}\nu x_{1}\cos(x_{3})\\\ +\alpha x_{1}x_{2}\end{array}$ | $\begin{array}[]{l}\alpha x_{1}^{2}\\\ -\nu x_{1}\sin(x_{3})\end{array}$ | $R(x_{3})$ | $\begin{array}[]{l}P_{1}+\alpha x_{2},\\\ P_{2}-\alpha x_{1},J_{3}+P_{3}\end{array}$ | $\tilde{R}(x_{3})$
13. | $\alpha x_{1}x_{2}$ | $\alpha x_{1}^{2}$ | $R(x_{3})$ | $\begin{array}[]{l}P_{1}+\alpha x_{2},Q,\tilde{Q}\\\ P_{2}-\alpha x_{1},\ J_{3}\end{array}$ | $\tilde{R}(x_{3})$
14. | $\alpha x_{1}x_{2}$ | $\alpha x_{1}^{2}$ | 0 | $\begin{array}[]{l}P_{1}+\alpha x_{2},P_{2}-\alpha x_{1},\\\ P_{3},G_{3},J_{3},Q,\tilde{Q}\end{array}$ | 0

Table 2. Generating functions and symmetries including dilatation

No | Functions $F$ | Functions $G$ | Potential $A^{0}$ | Symmetries | Potential $\tilde{A}$
---|---|---|---|---|---
1. | $\begin{array}[]{l}F(r^{2\alpha},r^{\alpha}\text{e}^{-\varphi})\\\ +\tilde{F}(\theta,r^{\alpha}\text{e}^{-\varphi})\end{array}$ | $G(\theta,r^{\alpha}\text{e}^{-\varphi})$ | $\frac{1}{r^{2}}{R(\theta,r^{\alpha}\text{e}^{-\varphi})}$ | $\begin{array}[]{l}D+\alpha J_{3}\end{array}$ | $\ln(r){\tilde{R}(\theta,r^{\alpha}\text{e}^{-\varphi})}$
2. | $\frac{1}{\tilde{r}}\tilde{F}(\varphi)$ | $G(\varphi)$ | $\frac{1}{\tilde{r}^{2}}R(\varphi)$ | $D,P_{3}$ | $\ln(r){\tilde{R}(\theta,r^{\alpha}\text{e}^{-\varphi})}$
3. | $F(\varphi,\theta)$ | $G(\varphi,\theta)$ | $\frac{1}{r^{2}}R(\varphi,\theta)$ | $A,\ D$ | $\ln(r){\tilde{R}(\theta,r^{\alpha}\text{e}^{-\varphi})}$
4. | $0$ | $\frac{\lambda}{r}x_{3}\varphi$ | $\frac{\mu}{r^{2}}$ | $\begin{array}[]{l}\frac{\lambda}{r}(x_{3}^{2}\partial_{2}\varphi-x_{1})+J_{1},\\\ \frac{\lambda}{r}(x_{2}+x_{3}^{2}\partial_{1}\varphi)-J_{2},\\\ D,\ A,J_{3}\end{array}$ | $\mu\ln({r})$
5. | $F(\theta)$ | $G(\theta)$ | $\frac{1}{r^{2}}R(\theta)$ | $A,\ D,\ J_{3}$ | $\ln(r)\tilde{R}(\theta)$
6. | $0$ | $G\left(\tilde{r}^{\alpha}\text{e}^{-\varphi}\right)$ | $\frac{1}{r^{2}}R\left(\tilde{r}^{\alpha}\text{e}^{-\varphi}\right)$ | $\begin{array}[]{l}D+\alpha L_{3},\\\ P_{3},\ G_{3}\end{array}$ | $\ln(r)\tilde{R}\left(\tilde{r}^{\alpha}\text{e}^{-\varphi}\right)$
7. | $F(\varphi)$ | $G(\varphi)$ | $\frac{1}{r^{2}}R(\varphi)$ | $\begin{array}[]{l}A,\ D,\\\ P_{3},\ G_{3}\end{array}$ | $\ln({r})\tilde{R}(\varphi)$
8. | $0$ | $\lambda\frac{x_{2}}{x_{3}}$ | $\frac{\mu}{x_{3}^{2}}$ | $\begin{array}[]{l}P_{1},\ P_{2},\ G_{2},\\\ D,\ A\end{array}$ | $\kappa\ln({x_{3}})$

Table 3. Generating functions and symmetries exponential in $t$.

No | Function $F$ | Function $G$ | Potential $A^{0}$ | Symmetries | Potentials $\tilde{A}$
---|---|---|---|---|---
1. | $0$ | $G(x_{1},x_{2})$ | $R(x_{1},x_{2})-\frac{\omega^{2}x_{3}^{2}}{2}$ | $\begin{array}[]{l}B^{+}_{3}(\omega),B^{-}_{3}(\omega)\end{array}$ | $\tilde{R}(x_{1},x_{2})$
2. | $0$ | $G(\tilde{r})$ | $R(\tilde{r})-\frac{\omega^{2}x_{3}^{2}}{2}$ | $\begin{array}[]{l}B^{+}_{3}(\omega),B^{-}_{3}(\omega),\ J_{3}\end{array}$ | $\tilde{R}(\tilde{r})$
3. | $0$ | $G(x_{1})$ | $R(x_{1})-\frac{\omega^{2}x_{3}^{2}}{2}$ | $\begin{array}[]{l}B^{+}_{3}(\omega),B^{-}_{3}(\omega),\\\ P_{2},\ G_{2}\end{array}$ | $\tilde{R}(x_{1})$
4. | $x_{1}F(x_{3})$ | $x_{1}G(x_{3})$ | $\begin{array}[]{l}R(x_{3})\\\ +\omega_{1}x_{1}F(x_{3})\\\ +\omega_{2}x_{2}G(x_{3})\\\ -\frac{\omega_{1}^{2}x_{1}^{2}}{2}-\frac{\omega_{2}^{2}x_{2}^{2}}{2}\end{array}$ | $\begin{array}[]{l}B^{+}_{1}(\omega_{1}),B^{-}_{1}(\omega_{1}),\\\ B^{+}_{2}(\omega_{2}),B^{-}_{2}(\omega_{2})\end{array}$ | $\tilde{R}(x_{3})$
5. | $F(x_{1},x_{2})$ | $G(x_{1},x_{2})$ | $\begin{array}[]{l}R({x_{1},x_{2}})-\frac{\omega^{2}x_{3}^{2}}{2}\\\ +\omega x_{3}F(x_{1},x_{2})\end{array}$ | $B^{+}_{3}(\omega)$ | $\tilde{R}(x_{1},x_{2})$
6. | $0$ | $G(\tilde{r})$ | $R({\tilde{r}})-\frac{\omega^{2}x_{3}^{2}}{2}+\kappa\varphi$ | $\begin{array}[]{l}B^{+}_{3}(\omega),B^{-}_{3}(\omega),\\\ L_{3}+\kappa t\end{array}$ | $\tilde{R}({\tilde{r}})$
7. | $0$ | $G(\tilde{r})$ | $\begin{array}[]{l}R({\tilde{r}})-\frac{\omega^{2}x_{3}^{2}}{2}\\\ +\omega x_{3}F(\tilde{r})+\kappa\varphi\end{array}$ | $\begin{array}[]{l}B^{+}_{3}(\omega),\\\ L_{3}+\kappa t\end{array}$ | $\tilde{R}({\tilde{r}})$
8. | $F(x_{1})$ | $G(x_{1})$ | $\begin{array}[]{l}R({x_{1}})-\frac{\omega^{2}x_{3}^{2}}{2}\\\ +\omega x_{3}F(x_{1})\end{array}$ | $\begin{array}[]{l}B^{+}_{3}(\omega),P_{2},\text{and }\\\ G_{2}\text{ if }G(x_{1})=0\end{array}$ | $\tilde{R}({x_{1}})$
9. | $\begin{array}[]{l}\frac{1}{r}F(\theta,\varphi)\end{array}$ | $G(\theta,\varphi)$ | $\begin{array}[]{l}\frac{1}{r^{2}}R(\theta,\varphi)-\frac{\omega^{2}r^{2}}{2}\\\ +\omega\partial_{\varphi}G(\theta,\varphi)\\\ +\omega\cos(\theta)F(\theta,\varphi)\end{array}$ | $A^{+}(\omega)$ | $\ln(r)\tilde{R}(\theta,\varphi)$
10. | $\begin{array}[]{l}\frac{1}{\tilde{r}}F(\varphi)\end{array}$ | $G(\varphi)$ | $\begin{array}[]{l}\frac{1}{r^{2}}R(\theta,\varphi)-\frac{\omega^{2}r^{2}}{2}\\\ +\frac{\omega x_{3}F(\varphi)}{\tilde{r}}\end{array}$ | $\begin{array}[]{l}B^{+}_{3}(\omega),A^{+}(\omega)\end{array}$ | $\ln(r)\tilde{R}(\theta,\varphi)$
11. | $\begin{array}[]{l}\frac{1}{r}F(\theta)\end{array}$ | $G(\theta)$ | $\begin{array}[]{l}\frac{1}{r^{2}}R(\theta)-\frac{\omega^{2}r^{2}}{2}\\\ +\omega\cos(\theta)F(\theta)\end{array}$ | $\begin{array}[]{l}A^{+}(\omega),J_{3}\end{array}$ | $\ln(r)\tilde{R}(\theta)$
12. | $\begin{array}[]{l}\frac{1}{\tilde{r}}\end{array}$ | $0$ | $\begin{array}[]{l}\frac{1}{\tilde{r}^{2}}R(\varphi)\\\ -\frac{\omega^{2}r^{2}}{2}+\frac{\omega x_{3}}{\tilde{r}}\end{array}$ | $\begin{array}[]{l}A^{+}(\omega),B^{+}_{3}(\omega)\end{array}$ | $\ln({\tilde{r}})\tilde{R}(\varphi)$
13. | $\begin{array}[]{l}\frac{\nu}{x_{1}}\end{array}$ | 0 | $-\frac{\omega^{2}r^{2}}{2}+\frac{\omega x_{3}}{x_{1}}$ | $\begin{array}[]{l}A^{+}(\omega),B_{2}^{+},B_{2}^{-},B^{+}_{3}(\omega)\end{array}$ | $0$
14. | $\begin{array}[]{l}x_{1}F(x_{3})\\\ +\alpha x_{1}x_{2}\end{array}$ | $\begin{array}[]{l}x_{1}G(x_{3})\\\ +\alpha x_{1}^{2}\end{array}$ | $\begin{array}[]{l}R(x_{3})-\frac{\omega^{2}\tilde{r}^{2}}{2}\\\ -2\alpha\omega x_{1}x_{2}\end{array}$ | $\begin{array}[]{l}B^{+}_{1}(\omega)+\alpha\text{e}^{\omega t}x_{2},\\\ B^{-}_{2}(\omega)-\alpha\text{e}^{-\omega t}x_{1}\end{array}$ | $\tilde{R}(x_{3})$
15. | $\alpha x_{1}x_{2}$ | $\alpha x_{1}^{2}$ | $-\frac{\omega^{2}x_{3}^{2}}{2}$ | $\begin{array}[]{l}P_{1}+\alpha x_{2},P_{2}-\alpha x_{1},\\\ B^{+}_{3}(\omega),B^{-}_{3}(\omega),L_{3},\\\ Q,\ \tilde{Q}\end{array}$ | $0$
16. | $\alpha x_{1}x_{2}$ | $\alpha x_{1}^{2}$ | $\begin{array}[]{l}-2\alpha\omega x_{1}x_{2}\\\ -\frac{\omega^{2}\tilde{r}^{2}}{2}\end{array}$ | $\begin{array}[]{l}B^{+}_{1}(\omega)+\alpha\text{e}^{\omega t}x_{2},\\\ B^{-}_{2}(\omega)-\alpha\text{e}^{-\omega t}x_{1},\\\ P_{3},G_{3},Q,\tilde{Q}\end{array}$ | $0$
17. | $\alpha x_{1}x_{2}$ | $\alpha x_{1}^{2}$ | $\begin{array}[]{l}-2\alpha\omega x_{1}x_{2}\\\ -\frac{\omega^{2}\tilde{r}^{2}}{2}-\frac{\omega_{3}^{2}x_{3}^{2}}{2}\end{array}$ | $\begin{array}[]{l}B^{+}_{1}(\omega)+\alpha\text{e}^{\omega t}x_{2},\\\ B^{-}_{2}(\omega)-\alpha\text{e}^{-\omega t}x_{1},\\\ B^{+}_{3}(\omega_{3}),B^{-}_{3}(\omega_{3}),\\\ Q,\tilde{Q}\end{array}$ | $0$

The classification results presented in Tables 1-3 are restricted to the
systems with non-trivial magnetic fields and non-trivial Pauli terms. The
complete list of the inequivalent systems with trivial magnetic field can be
found in paper [28]. For the readers convenience and for our future needs we
present the remaining systems with trivial vector potentials in Table 4 where
the last item is needed only for the more general equation considered in the
following sections.

Table 4. Symmetries of systems with pure electric field.

No | Potential $A^{0}$ | Symmetries | Potentials $\tilde{A}$
---|---|---|---
1. | $\begin{array}[]{l}\frac{1}{\tilde{r}^{2}}R(\varphi)\end{array}$ | $\begin{array}[]{l}D,A,P_{3},G_{3}\end{array}$ | $\ln(\tilde{r})\tilde{R}(\varphi)$
2. | $\begin{array}[]{l}\frac{\kappa}{\tilde{r}^{2}}\end{array}$ | $\begin{array}[]{l}D,A,J_{3},P_{3},G_{3}\end{array}$ | ${\mu}\ln({\tilde{r}})$
3. | $\begin{array}[]{l}\frac{\kappa}{x_{2}^{2}}\end{array}$ | $\begin{array}[]{l}A,D,P_{1},P_{3},\\\ G_{1},G_{3},J_{2}\end{array}$ | $\begin{array}[]{l}\mu\ln({x_{2}})\end{array}$
4. |  $\begin{array}[]{l}R(r)\end{array}$ | $\begin{array}[]{l}J_{1},J_{2},J_{3}\end{array}$ | $\tilde{R}(r)$
5. | $\begin{array}[]{l}R(x_{3})\end{array}$ | $\begin{array}[]{l}P_{1},P_{2},G_{1},G_{2},J_{3}\end{array}$ | $\tilde{x}R(x_{3})$
6. | $\begin{array}[]{l}R(x_{3})-\frac{\omega^{2}\tilde{r}^{2}}{2}\end{array}$ | $\begin{array}[]{l}B^{+}_{1}(\omega_{1}),B^{-}_{1}(\omega_{1}),\\\ B^{+}_{2}(\omega_{2}),B^{-}_{2}(\omega_{2}),J_{3}\end{array}$ | $\tilde{R}(x_{3})$
7. | $\begin{array}[]{l}-\frac{\omega_{1}^{2}x_{1}^{2}}{2}-\frac{\omega_{2}^{2}x_{2}^{2}}{2}\end{array}$ | $\begin{array}[]{l}B^{+}_{1}(\omega_{1}),B^{-}_{1}(\omega_{1}),P_{3},\\\ B^{+}_{2}(\omega_{2}),B^{-}_{2}(\omega_{2}),G_{3}\end{array}$ |  0
8. | $\begin{array}[]{l}-\frac{\omega_{1}^{2}x_{1}^{2}}{2}-\frac{\omega_{2}^{2}x_{2}^{2}}{2}-\frac{\omega_{3}^{2}x_{3}^{2}}{2}\end{array}$ | $\begin{array}[]{l}B^{+}_{1}(\omega_{1}),B^{-}_{1}(\omega_{1}),\\\ B^{+}_{2}(\omega_{2}),B^{-}_{2}(\omega_{2}),\\\ B^{+}_{3}(\omega_{3}),B^{-}_{3}(\omega_{3})\end{array}$ |  0
9. | $\begin{array}[]{l}-\frac{\omega^{2}x_{3}^{2}}{2}\end{array}$ | $\begin{array}[]{l}P_{1},P_{2},G_{1},G_{2}J_{3}\\\ B^{+}_{3}(\omega),\ B^{-}_{3}(\omega),\end{array}$ |  0
10. | $\begin{array}[]{l}-\frac{\omega^{2}\tilde{r}^{2}}{2}\end{array}$ | $\begin{array}[]{l}B^{+}_{1}(\omega_{1}),B^{-}_{1}(\omega_{1}),\\\ B^{+}_{2}(\omega_{2}),B^{-}_{2}(\omega_{2}),\\\ J_{3},P_{3},G_{3}\end{array}$ |  0
11. | $\begin{array}[]{l}-\frac{\omega^{2}\tilde{r}^{2}}{2}-\frac{\omega_{3}^{2}x_{3}^{2}}{2}\end{array}$ | $\begin{array}[]{l}B^{+}_{1}(\omega_{1}),B^{-}_{1}(\omega_{1}),\\\ B^{+}_{2}(\omega_{2}),B^{-}_{2}(\omega_{2}),\\\ B^{+}_{3}(\omega_{3}),B^{-}_{3}(\omega_{3}),J_{3}\end{array}$ |  0
12. | $\begin{array}[]{l}\frac{\kappa}{r^{2}}\end{array}$ | $\begin{array}[]{l}A,D,J_{1},J_{2},J_{3},\\\ {\text{ and }}\hat{Q}{\text{ for}}\\\ {\text{ Hamiltonian }}(\ref{H3333})\end{array}$ | $\mu\ln({r})$
13. |  $\begin{array}[]{l}\frac{1}{2\nu}\ln r\end{array}$ | $\begin{array}[]{l}J_{1},J_{2},J_{3},\\\ {\text{ and }}\hat{Q}{\text{ for}}\\\ {\text{ Hamiltonian }}(\ref{H3333})\end{array}$ | 0

Thus we complete the classification of symmetries of SP equations with time
independent potentials. Such equations keep all symmetries of the SEs with
scalar and vector potentials (which, however, should be slate modified by
changing orbital momenta by the total orbital momenta) and can admit
additional matrix symmetries. Notice that the analogous result has been proven
earlier for the stationary Schrödinger equation [37].

##  3 Quasi relativistic Schrödinger equation

It is pretty well known that SP equation is nothing but the first
approximation of the relativistic Dirac equation provided the approximation
parameter is the inverse speed of light $\frac{1}{c}$. Moreover, the Dirac
equation predicts the fixed and correct value of the coupling constant $g$ in
equation (2).

Considering the next approximation (which can be found starting with the Dirac
equation and applying the Foldy-Wouthuysen transformation in the approximate
[38] or exact [39] forms) we come to the quasi relativistic Schrödinger
equation (QRSE), which, in addition to Pauli term, includes two additional
terms describing the spin-orbit and Darwin couplings. The corresponding
Hamiltonian being written in vector notations looks as follows:

$\displaystyle
H=\frac{{\mathbf{\pi}}^{2}}{2m}+neA^{0}-\frac{ne\hbar}{2mc}{\mbox{\boldmath$\sigma$\unboldmath}}\cdot{\bf
H}-\frac{ne\hbar}{8m^{2}c^{2}}{\mbox{\boldmath$\sigma$\unboldmath}}\cdot({\bf
E}\times{\mbox{\boldmath$\pi$\unboldmath}}-{\mbox{\boldmath$\pi$\unboldmath}}\times{\bf
E})+\frac{ne\hbar}{8m^{2}c^{2}}div{\bf E}$ (59)

where $m$ is a particle mass, $e$ is the electron charge, $\hbar$ is Plank
constant, $c$ is the speed of light, ${\bf E}=\nabla A^{0}$ and
$\pi^{a}=p^{a}-\frac{ne}{c}A^{a}$.

Notice that in atomic units $\hbar=e=1,c=137$. In this units the electronic
mass is normalized to the unity also. But we keep the mass term arbitrary to
reserve the possibility to describe a motion of a particle whose mass differs
from the electron one. In addition, the described particle can have an
arbitrary charge $ne$ with a positive integer $n$.

In (59) we also keep the terms proportional to $\frac{n^{2}e^{2}}{c^{3}}$
which can be essential for large $n$. For $n=1$ it is reasonable to omit them
and reduce the considered Hamiltonian to the following form:

$\displaystyle
H=\frac{{\mathbf{p}}^{2}}{2m}+eA^{0}-\frac{e\hbar}{2mc}{\mbox{\boldmath$\sigma$\unboldmath}}\cdot{\bf
H}-\frac{e\hbar}{8m^{2}c^{2}}{\mbox{\boldmath$\sigma$\unboldmath}}\cdot({\bf
E}\times{\bf p}-{\bf p}\times{\bf E})+\frac{e\hbar}{8m^{2}c^{2}}div{\bf E}.$
(60)

The first three terms in the r.h.s. of equation (60) represent the SP
Hamiltonian for neutral particles whose symmetries were classified in recent
paper [29]. Surely this classification can be extended to the case of systems
with spin-orbit couplings, but we will not do it here since it would need an
enormous extension of the paper size.

We will consider also a more general quasi relativistic Hamiltonian which
includes the additional potential $S({\bf x})$:

$\displaystyle
H=\frac{{\mathbf{\pi}}^{2}}{2m}+A^{0}-\frac{\hbar}{2mc}{\bf{\sigma}}\cdot{\bf
H}-\frac{\hbar}{8m^{2}c^{2}}{\mbox{\boldmath$\sigma$\unboldmath}}\cdot({\tilde{\bf
E}}\times{\mbox{\boldmath$\pi$\unboldmath}}-{\mbox{\boldmath$\pi$\unboldmath}}\times{\tilde{\bf
E}})+\frac{\hbar}{8m^{2}c^{2}}div{\bf{\tilde{E}}}$ (61)

where

$\displaystyle\tilde{\bf E}=\nabla\tilde{A},\quad\tilde{A}=neA^{0}-qS({\bf
x}).$ (62)

In contrast with (59) the electric field is replaced by the generalised
effective vector field $\tilde{E}$ which is the gradient of the another
potential $\tilde{A}$. In other words now we have the additional scalar
potential which has clear relativistic roots. Namely, just Hamiltonian (61)
corresponds to the quasi relativistic approximation of the Dirac equation
which includes the superposition of the four-vector and scalar potentials. Of
course this Hamiltonian is reduced to (59) in the particular case when the
coupling constant $q$ is trivial.

We will search for symmetries of QRSE which includes the most general
Hamiltonian (61) and its reduced version (59). First let us rewrite (59) in a
more simple form using tensor notations and renormalized independent variables
and potentials:

$\displaystyle
H=\frac{1}{2}\pi_{a}\pi_{a}+A^{0}+g\sigma_{a}H^{a}+\frac{\nu}{2}\varepsilon_{abc}\sigma_{a}(E^{b}p^{c}+p^{c}E^{b})+\mu\Delta
A^{0}$ (63)

where $\Delta$ is the Laplace operator, $\mu$ and $\nu$ are the coupling
constants which are proportional to the squared inverse speed of light
$\frac{1}{c^{2}}$ and $E^{b}$ are components of the electric strength vector
which in our case are proportional to $\partial_{b}A^{0}$. The constants $\mu$
and $\nu$ can be treated as rescaled multipliers presented (59), but we will
not fix them and treat as arbitrary ones, which can appear as an effect of the
anomalous interaction. Notice that Hamiltonian (60) also can be rewritten in
the form analogous to (63) but with $\pi^{a}$ changed by $p^{a}$.

The more general Hamiltonian (61) also can be written in tensor notations:

$\displaystyle
H=\frac{1}{2}\pi_{a}\pi_{a}+\tilde{A}+g\sigma_{a}H^{a}+\frac{\nu}{2}\varepsilon_{abc}\sigma_{a}(\tilde{E}^{b}\pi^{c}+\pi^{c}\tilde{E}^{b})+\mu(\partial_{a}\tilde{E}_{a})$
(64)

with potentials and components of the electric field are defined in (62).

The SEs with Hamiltonians (63) and (64) are invariant with respect to neither
Poincaré nor Galilei group. At the best of my knowledge symmetry of this
equation with respect to Lie groups is still terra incognita. In the following
sections we will search for these symmetries.

###  3.1 Generalized determining equations

In Section 2 we have solved the subproblem of our classification problem which
corresponds to the trivial coupling constants $\lambda$ and $\nu$, which is a
particular but very important part of our task. In the present section we
consider the general case with non-trivial spin-orbit and Darwin terms.

The specificity of the considered classification problems is that the free
elements $\lambda$ and $\nu$ are not dimensionless but should be proportional
to the squared inverse speed (of light), and so it can be changed under the
dilatation transformations. However in the atomic units the speed of light is
reduced to the dimensionless constant c=137 and we shall treat it as a non
changing unit which commutes with symmetry group generators.

Substituting Hamiltonian (63) into the operator equation (11) we again obtain
the determining equations (12)-(14). However, the remaining equations (15),
(16) will include the additional terms proportional to $\lambda$ and $\nu$
which can be indicated in the following formulae:

$\displaystyle\begin{split}&\dot{\xi}^{a}+\frac{1}{2}\eta_{a}=e(A^{b}\xi^{a}_{b}-\xi^{b}A^{a}_{b}+\alpha
A^{a})\\\
&+\frac{\nu}{2}(\varepsilon_{bcd}\sigma^{c}A^{0}_{d}\xi^{a}_{b}-\varepsilon_{acd}\sigma^{c}(\xi^{k}A^{0}_{kd}+\alpha
A^{0}_{d})-\frac{1}{2}\varepsilon_{acd}[\sigma^{c}A^{0}_{d},\eta]),\end{split}$
(65) $\displaystyle\xi^{a}V_{a}=\alpha
V+\dot{\eta}+{\mathrm{i}}[\eta,V]-A^{a}\eta_{a}-\frac{\nu}{2}\varepsilon_{acd}\\{\sigma^{c}A^{0}_{d},\eta_{a}\\}$
(66)

where $V$ is the modified matrix potential

$\displaystyle\begin{split}&V=V^{0}+g\sigma_{a}\tilde{H}^{a},\quad\tilde{H}^{a}=H^{a}-2\nu\varepsilon^{abc}A^{b}E^{c},\\\
&V^{0}=A^{0}+\frac{1}{2}((A^{1})^{2}+(A^{2})^{2}+(A^{3})^{2})+\mu\Delta
A^{0}.\end{split}$ (67)

.

By construction equations (65) and (66) include $2\times 2$ matrices, i.e., in
fact any of them is nothing but a system of four equations. Collecting in (65)
the terms proportional to the unit matrix we again come to equation (17). The
terms proportional to matrix $\sigma^{c}$ look as follows:

$\displaystyle\eta^{c}_{a}=\nu(\varepsilon_{bcd}A^{0}_{d}\xi^{a}_{b}-\varepsilon_{acd}(\xi^{k}A^{0}_{kd}+\alpha
A^{0}_{d})+2\delta^{ac}\eta^{k}A^{0}_{k}-2\eta^{a}A^{0}_{c})$ (68)

where we again use the expansion (10). In contrast with (18) the r.h.s. of
equation (68) is not trivial.

Equation (66) also can be decoupled to the scalar and vector parts:

$\displaystyle\xi^{a}A^{0}_{a}=\alpha
V^{0}+\dot{\eta}^{0}-A^{a}\eta^{0}_{a}-\nu\varepsilon_{acd}\eta^{c}_{a}A^{0}_{d}$
(69)

and

$\displaystyle
g(\xi^{a}\tilde{H}_{a}^{b}-\alpha\tilde{H}^{b}+2\varepsilon^{bcd}\eta^{c}\tilde{H}^{d})-\dot{\eta}^{b}+\nu\varepsilon_{abd}\eta^{0}_{a}A^{0}_{d}-\eta^{b}_{a}A^{a}=0.$
(70)

Substituting (17) into (69) and using definition (67) we reduce the latter
equation to the following form:

$\displaystyle\xi^{a}(A^{0}_{a}+\lambda\Delta
A^{0}_{a})+\dot{\xi}^{a}A^{a}=\alpha(A^{0}+\lambda\Delta
A^{0})+\dot{\eta}^{0}-\nu\varepsilon_{acd}\eta^{c}_{a}A^{0}_{d}.$ (71)

Finally, substituting the expression (68) for $\eta^{a}_{c}$ into (71) we
obtain:

$\displaystyle\begin{split}&\hskip 34.1433pt\xi^{a}(A^{0}_{a}+\lambda\Delta
A^{0}_{a})+\dot{\xi}^{a}A^{a}=\alpha(A^{0}+\lambda\Delta
A^{0})+\dot{\eta}^{0}\\\ &+\nu^{2}(2\alpha
A^{0}_{d}A^{0}_{d}+2\xi^{k}A^{0}_{kd}A^{0}_{d}+\xi^{n}_{n}A^{0}_{m}A^{0}_{m}-\xi^{m}_{n}A^{0}_{m}A^{0}_{n}).\end{split}$
(72)

Notice that for Hamiltonian (64) equations (68), (70) and (72) are generalised
to the following form:

$\displaystyle\eta^{c}_{a}=\nu(\varepsilon_{bcd}\tilde{A}_{d}\xi^{a}_{b}-\varepsilon_{acd}(\xi^{k}\tilde{A}^{k}_{d}+\kappa\alpha\tilde{A}_{d})+2\delta^{ac}\eta^{k}\tilde{A}_{k}-2\eta^{a}\tilde{A}_{c}),$
(73) $\displaystyle
g(\xi^{a}\tilde{H}_{a}^{b}-\alpha\tilde{H}^{b}+2\varepsilon^{bcd}\eta^{c}\tilde{H}^{d})-\dot{\eta}^{b}+\nu\varepsilon_{abd}\eta^{0}_{a}A^{0}_{d}-\eta^{b}_{a}A^{a}=0$
(74)

and

$\displaystyle\xi^{a}(\tilde{A}_{a}+\lambda\Delta\tilde{A}_{a})+\dot{\xi}^{a}A^{a}=\alpha(\tilde{A}+\lambda\Delta
A^{0})+\dot{\eta}^{0}-\nu\varepsilon_{acd}\eta^{c}_{a}\tilde{A}_{d}$ (75)

where $\tilde{H}^{a}=H^{a}-2\nu\varepsilon^{abc}A^{b}\tilde{A}_{c}.$

Thus to describe Lie symmetries of the QRSE with Hamiltonian (63) we are
supposed to solve the system of determining equations (12)-(14), (17), (68),
(70) and (72), while for Hamiltonian (64) we have the system (12)-(14), (17),
(73), (74) and (75). In spite of its frightening view these systems can be
effectively solved, since the solution of the main part of them have been
obtained in the previous section.

###  3.2 Equivalence relations

Any search for Lie symmetries of a partial differential equation should
include a consistent definition of admissible (equivalence) transformations
which keep the generic form of this equation but can change its arbitrary
elements (in our case potentials $A^{0}$, $\tilde{A}$ and $A^{a})$.

Till now we restricted ourselves to the direct use the equivalence gruppoid
for the Schrödinger equation found in paper [28] since in fact all
inequivalent potentials are fixed by the determining equations (12), (13),
(14), (17) and (19) which are the same as in the case of trivial Pauli, spin-
orbit and Darwin terms. A particular sector of the equivalence relations is
formed by gauge transformations which change vector-potentials but keep the
vectors of electric and magnetic fields invariant.

But whenever the spin-orbit coupling is nontrivial, we can find a new effect
the gauge transforms which consists in changing the magnetic field present in
the Pauli term. Indeed, changing $\psi\to\exp(i\Phi)\psi$ in the generalised
Schrödinger equation including Hamiltonian (63) we observe the following
change of the Pauli term:

$\displaystyle g\sigma^{a}H^{a}\to g\sigma^{a}H^{\prime a}$ (76)

where

$\displaystyle H^{\prime
a}=H^{a}-\frac{\nu}{g}\varepsilon^{abc}\Phi_{b}E^{c}.$ (77)

In particular, for some class of the external electric and magnetic fields the
related Pauli term can be reduced to zero. The necessary property satisfied by
such fields is that $\bf E$ and $\bf H$ are orthogonal, i.e., $E^{a}H^{a}=0$.
In other words, one of the invariants of the electromagnetic field should be
trivial.

Thus the gauge transformations being applied to the QRSE with Hamiltonians
(63), (59) and (60) can change the effective magnetic field, and this effect
is rather specific and cannot be observed in Schrödinger, Schrödinger-Pauli
and Dirac equations. Important examples of the magnetic field which can be
nullified in Hamiltonian (63) by the charge transformations are the constant
magnetic field and the field generated by the infinite straight current.

###  3.3 Solution of the determining equations

Since we have in hands all inequivalent solutions of equations (12)-(14) and
(17) which were found in paper [28], we have a priori information about
inequivalent versions of vector potentials and symmetries which are
represented in Tables 1–3. Thus the only thing we need is to verify whether
these vector potentials are compatible with equations (68), (70), (72) for
Hamiltonians (63) and equations (73), (74), (75) for Hamiltonians (64). In
addition it is necessary to study matrix integrals of motion which do not
include differentials, since they are not fixed by the subsystem of the
determining equations

Like in Section 2 we consider vector potentials which correspond to
inequivalent one dimension symmetry algebras. For symmetry presented in (45)
equations determining equations (17), (68), (70) and (72) look as follows:

$\displaystyle\eta^{0}_{a}=-A^{a}_{3},$ (78)
$\displaystyle\eta^{c}_{a}=\nu(-\varepsilon_{acd}A^{0}_{3d}+2\delta^{ac}\eta^{k}A^{0}_{k}-2\eta^{a}A^{0}_{c}),$
(79) $\displaystyle
g(\tilde{H}_{3}^{b}+2\varepsilon^{bcd}\eta^{c}\tilde{H}^{d})-\dot{\eta}^{b}+\nu\varepsilon_{abd}\eta^{0}_{a}A^{0}_{d}-\eta^{b}_{a}A^{a}=0$
(80)

and

$\displaystyle(A^{0}+\mu\Delta
A^{0}-{\nu^{2}}A^{0}_{d}A^{0}_{d})_{3}=\dot{\eta}^{0}.$ (81)

It was shown in paper [28] that up to gauge transformations the generic
solution of the scalar equation (78) is given by equations (5) and (22) while
the related function $\dot{\eta}^{0}$ is trivial.

Using the identity

$\displaystyle\varepsilon_{abc}(\eta^{c}_{a})_{b}=0$ (82)

we obtain the following differential consequence of equations (79):

$\displaystyle(\Delta A^{0}-\nu A^{0}_{a}A^{0}_{a})_{3}=0$ (83)

which is compatible with (81) (where $\dot{\eta}^{0}$ is trivial ) only for
the case $A^{0}_{3}=0$. The corresponding functions $\eta^{k}$ solving
equations (79) and (80) are trivial also, and we again come to the generating
functions and symmetries fixed in Item 1 of table 1.

We see that if the SP equation (2), (7) admits symmetry $P_{3}$ then this
symmetry is admitted by the more general equation (63) with non-trivial
parameters $\nu$ and $\lambda$. Moreover, this symmetry cannot be generalized
by adding a matrix term, i.e., a more general motion constant (45) can be
present only with trivial functions $\eta^{\mu}$.

The next system we consider is given by relations (26). The corresponding
symmetry (27) again has to be generalised by adding matrix $\eta$ and
represented in the form (48). Moreover, the components $\eta^{a}$ should
satisfy condition (68). It is possible to verify by direct calculation that
the related equations (68), (70) and (71) are satisfied provided
$\eta^{1}=\eta^{2}=0,\eta^{3}=\frac{1}{2}$. We will not reproduce these
calculations since this result in fact is evident: the spin-orbit term is a
scalar with respect to rotations since the related electric field is a vector
as like as
${\mbox{\boldmath$\sigma$\unboldmath}}=(\sigma^{1},\sigma^{2},\sigma^{3})$ and
${\bf p}=(p^{1},p^{2},p^{3})$.

Consider now the systems whose scalar parts are given by equation (30). The
related generator (31) should be generalized to the following form:

$\displaystyle Q=L_{3}+p_{3}+\eta^{0}+\sigma_{a}\eta^{a}$ (84)

and so the corresponding determining equations (17), (68), (69) and (70) look
as follows:

$\displaystyle\eta^{0}_{a}=\varepsilon^{3ab}A^{b}-A^{a}_{\kappa},$ (85)
$\displaystyle\eta^{c}_{a}=\nu(\delta^{3c}A^{0}_{a}-\delta^{ac}A^{0}_{3}-\varepsilon_{acd}A^{0}_{d\kappa}+2\delta^{ac}\eta^{k}A^{0}_{k}-2\eta^{a}A^{0}_{c}),$
(86) $\displaystyle
g(\tilde{H}_{\kappa}^{b}+2\varepsilon^{bcd}\eta^{c}\tilde{H}^{d})-\dot{\eta}^{b}+\nu\varepsilon_{abd}\eta^{0}_{a}A^{0}_{d}-\eta^{b}_{a}A^{a}=0,$
(87) $\displaystyle(A^{0}+\lambda\Delta
A^{0}-{\nu^{2}}A^{0}_{d}A^{0}_{d})_{\kappa}=\dot{\eta}^{0}$ (88)

where $A^{0}_{\kappa}=\partial_{\kappa}A^{0},\
\tilde{H}^{b}_{\kappa}=\partial_{\kappa}\tilde{H}^{b},$ and
$\kappa=\arctan\frac{x_{2}}{x_{1}}+x_{3}.$

It is possible to show that up to gauge transformations the generic solution
of equation (85) can be presented in the form given in Item 3 of Table 1, see
paper [28]. The related function $\dot{\eta}^{0}$ is trivial.

The differential consequence (82) for vector equations (86) takes the
following form:

$\displaystyle(\Delta A^{0}-\nu A^{0}_{a}A^{0}_{a})_{\kappa}=0$ (89)

which is compatible with (87) only for the case $A^{0}_{\kappa}=0$. The
corresponding functions $\eta^{k}$ solving equations (79) and (80) are
trivial, and we came to the generating functions and symmetries fixed in Item
3 of Table 1.

The symmetries discussed in the above are valid for both Schrödinger-Pauli and
quasirelativistic Hamiltonians. In contrary, symmetries of the system
specified by relations (34) are valid for SP equation but not for QRSE with
Hamiltonian (63) .

To prove this facts it is sufficient to consider the related determining
equations (17 ), (68), (70) and (72) which are reduced to the following form:

$\displaystyle\begin{split}&\eta^{0}_{1}=A^{1}+\mu(A^{1}+2A^{1}_{y}),\ \
\eta^{0}_{2}=A^{2}-\mu(A^{1}+2A^{2}_{y}),\quad\eta^{0}_{3}=A^{3},\end{split}$
(90)
$\displaystyle\eta^{c}_{a}=\nu(\varepsilon_{acd}((1-\kappa)A^{0}_{d}-A^{0}_{dy})-2\eta^{a}A^{0}_{c}+2\delta^{ac}\eta^{b}A^{0}_{b}+\mu\delta^{3c}A^{0}_{a}),$
(91) $\displaystyle
g(\tilde{H}^{b}_{y}-\alpha\tilde{H}^{b}+2\varepsilon^{bcd}\eta^{c}\tilde{H}^{d})-\dot{\eta}^{b}+\nu\varepsilon^{abd}\eta^{0}_{a}A^{0}_{d}=0,$
(92)
$\displaystyle(1+\mu\Delta)A^{0}_{y}=\alpha(1+\kappa\Delta)A^{0}+\dot{\eta}^{0}+\nu^{2}(2(1+\kappa\alpha)A^{0}_{d}A^{0}_{d}+(A^{0}_{d}A^{0}_{d})_{y})$
(93)

where $y=\mu\ln\tilde{r}+\varphi$.

In accordance with [28] the generic solution of equations (90) can be
represented in the form given in Item 4 of Table 1, and the related function
$\dot{\eta}^{0}$ present in equation (93) is trivial. Considering again the
differential consequence (82) of equation (91) and comparing it with (93)
conclude that the system (90)–(93) is incompatible.

Considering in the analogous manner the potentials and symmetry fixed in (38)
and (39) we come to the following version of the determining equations (17),
(68), (70) and (72):

$\displaystyle\eta^{0}_{a}=-\exp(\omega t)(\omega\delta_{a3}+A^{a}_{3})$ (94)
$\displaystyle\eta^{c}_{a}=\nu(-\varepsilon_{acd}A^{0}_{3d}\exp(\omega
t)+2\delta^{ac}\eta^{k}A^{0}_{k}-2\eta^{a}A^{0}_{c})$ (95) $\displaystyle
g(\exp(\omega
t)\tilde{H}_{3}^{b}+2\varepsilon^{bcd}\eta^{c}\tilde{H}^{d})-\dot{\eta}^{b}+\nu\varepsilon_{abd}\eta^{0}_{a}A^{0}_{d}-\eta^{b}_{a}A^{a}=0$
(96)

and

$\displaystyle(A^{0}+\mu\Delta
A^{0}-{\nu^{2}}A^{0}_{d}A^{0}_{d})_{3}=\dot{\eta}^{0}.$ (97)

It was shown in [28] that up to gauge transformations the generic solution of
(94) is given by equation (22) while the related function $\dot{\eta}^{0}$ is
equal to $\omega x_{3}$. Considering the differential consequence (82) of
relations (95) and comparing it with (97) we conclude that these equations are
incompatible, and so symmetry (39) generalised by adding a matrix $\eta$ is
not accepted by the QRSE with Hamiltonian (59). The same is true for symmetry
(43), and to prove this fact it is sufficient to repeat the above
manipulations staring with equation (94).

Thus we specify all inequivalent one dimensional symmetry algebras which can
be admitted by the QRSE with Hamiltonian (63). It happens that symmetries
(45), (48) and (84) are valid provided the potentials have the form presented
in Items 1, 2 and 3 of Table 1. The same is true for linear combinations of
these symmetries specified in the remaining items of this table. In contrary,
the symmetries (35), (39) and (43) are not valid.

###  3.4 Matrix integral of motion

The main result of the previous section can be formulated as follows: Lie
symmetries of the SP equation presented in Table 1 are kept also for the QRSE.
The only exceptions are the matrix symmetries (55) the first of which cannot
be kept for any spin-orbit coupling term while the second one is possible if
the scalar potential satisfies some additional restrictions.

In this section we find matrix symmetries for the QRSE. The generic form of
the corresponding symmetry operators again is given by formulae (9) and (10)
where, however, functions $\xi^{0}$ and $\xi^{a}$ are trivial. In this case
the determining equations (17), (68), (70) and (72) are reduced to the
following ones:

$\displaystyle\eta^{0}_{a}=0,\quad\dot{\eta}^{0}=0,$ (98)
$\displaystyle\dot{\eta}^{a}=2g(x_{b}A^{b})_{a}-(2g+2g\nu+1)A^{a}+\frac{x_{a}x_{b}A^{b}}{r^{2}}-x_{b}A^{a}_{b},$
(99)
$\displaystyle\eta^{c}_{a}=2\nu(\delta^{ac}\eta^{b}A^{0}_{b}-2g\eta^{a}A^{0}_{c}).$
(100)

Equation (100) is solved by the following functions:

$\displaystyle A^{0}=\frac{1}{2\nu}\ln(r),\quad\eta^{a}=\phi\frac{x_{a}}{r}$
(101)

where $\phi$ is a function of $t$ which is reduced to a constant if equation
(99) is satisfied. .

Thus we discover the following integral of motion:

$\displaystyle\hat{Q}=\frac{\sigma^{a}x_{a}}{r}$ (102)

which commute with Hamiltonian (63) provided scalar potential $A^{0}$ is given
by equation (101) and vector potential $A^{a}$ satisfies condition (99) with
$\eta^{a}$ given by equation (101). The generic form of the vector-potential
satisfying this condition is given by the following equation:

$\displaystyle
A^{a}=\frac{\varepsilon^{abc}\phi^{b}x_{c}}{r^{1+\nu+\frac{1}{g}}},\ g\neq 0$
(103)

where $\phi^{b},b=1,2,3$ are arbitrary functions of angular variables
$\varphi$ and $\theta$.

In particular vector potential can be trivial, then the corresponding QRSE
with scalar potential (101) admits symmetry (102) and three additional
symmetries

$\displaystyle J^{1}=L^{1}+\frac{1}{2}\sigma_{1},\quad
J^{2}=L^{2}+\frac{1}{2}\sigma_{2},\quad J^{3}=L^{3}+\frac{1}{2}\sigma_{3}$
(104)

where $L_{1},L_{2}$ and $L_{3}$ are components of the orbital momentum given
by equation (56). For nontrivial vector potentials represented in (104) this
set of symmetries is missing. However if functions $\phi^{a}$ are constant
there is an additional integral of motion, namely, a linear combination of
$J^{a}$. In this case up to rotation transformation we can set
$\xi^{1}=\xi^{2}=0,\xi^{3}=const$, then the additional integral of motion is
reduced to $J^{3}$.

##  4 Summary on symmetries of QRSE

In the previous sections we specify Lie symmetries of the SP equation which
are valid also for the quasi relativistic Schrödinder equation. In particular,
the complete list of potentials generating non-equivalent symmetries for QRSE
with Hamiltonian (63) have been obtained and fixed in Table 1. Notice that the
symmetries presented in this table are valid also for the SP equation, and the
addition spin-orbit term presented in the generalized Hamiltonian (63) keeps
all of them. However, the symmetries of SP equation presented in Tables 2–4
are lost provided the spin-orbit coupling be introduced.

Let us stress that the more general QRSE with Hamiltonian (64) keeps all
symmetries of the SP equation provided the potential $\tilde{A}$ has some
special forms. Moreover, for the special potentials presented in Items 12 and
13 of Table 4 we have the additional matrix symmetries which are missed in the
case of SP equation. To prove these facts it is sufficient to repeat the
calculations presented in Section 3.3 for a bit more general system of the
determining equations (73), (74) and (75) which are rather similar to (68),
(70) and (72) but includes the additional arbitrary element $\tilde{A}$. We
will not present here the calculation details but restrict ourselves to the
presentation of their results in the right columns of Tables 2–4.

##  5 Discussion

We present the completed group classification of Schrödinger-Pauli and the
quasi relativistic SEs. In other words we create certain group-theoretical
grounds for all quasirelativistic quantum mechanical models of the spin-orbit
and Pauli couplings. The main result of our research consists in obtaining the
a priori information about all inequivalent Lie symmetries which can be
accepted by such models. Moreover, this information is constructive since all
inequivalent potentials of the external electromagnetic field involved into
such models are specified also.

The presented results can be treated as a direct extension of the basic
symmetries of the free Schrödinger equation [24, 25] to the case of more
general equations including potentials and the above mentioned coupling terms.
In some sense these results are complete since we consider almost all possible
relativistic correction of order $1/c^{2}$, and it is well known that the
corrections of higher order in the inverse speed of light are not reasonable
for the one particle SEs. The only term of order $1/c^{2}$ which we ignore is
the relativistic correction to the kinetic energy $p^{4}/8m^{3}$. Its presence
reduces the number of symmetries admitted by the quasi relativistic equation,
but not too dramatically. Namely, symmetries presented in Table 1 are kept if
this term is added, but the generators of pure Galilei transformation $G_{2}$
and $G_{3}$ should be excluded. The same is true for the systems fixed in
Items 1–3 of Table 4. However, symmetries presented in Tables 2 and 3 and
Items 4–13 of Table 4 are not valid in presence of the relativistic correction
of the kinetic energy.

Notice that symmetries of the SEs with the relativistic correction of the
kinetic energy were discussed in old paper [40].

Let us remind that the Pauli term in Hamiltonians (6) and (63) is not
necessary to be interpreted as a relativistic correction, since it is
naturally generated (with the correct coupling constant) by the Galilei
invariant Levi-Leblond equation [41]. Moreover, the spin-orbit and Darwin
terms in (63) also can be obtained starting with first order Galilei invariant
motion equations [42]. But in contrast with the Dirac equation, Galilei
invariant ones do not predict a fixed value of the related coupling constants.

Like in papers [27, 28] to solve our classification problem we apply the
algebraic approach whose main idea is the systematic use of the subgroup
structure of the equivalence group. An important element of our analysis was
the fixation of the equivalence qroupoid of the analysed class of equations.
It happens that this gruppoid has the same structure as in the case of the
standard Schrödinger equation with arbitrary scalar and vector potentials,
thus we could simple use the results of paper [28] which form a ground for the
study of more general equations. We do not present the equivalence relations
here since for the sets of potentials presented in Tables 1-4 these relations
are fixed in [28]. The only potentials missing in [28] are given by equations
(100) and (101), but the corresponding equivalence relations belong to the
symmetry group of the QRSE or are realized by the gage transformations.

The latter potentials correspond to the system with a rather exotic integral
of motion being a matrix dependent on spatial variables which is fixed by
equation (102). Notice that in the case when the vector potential $A^{a}$ is
trivial this system appears to be maximally superintegrable, since, in additon
to the symmetries indicated in Item 2 of Table 4 there exist one more matrix
integral of motion

$\displaystyle\tilde{Q}=\sigma^{a}L^{a}+1.$ (105)

Operators (102) and (105) commute with the Hamiltonian and total orbital
momentum, but anticommute between themselves. Thus we have a perfect example
of symmetry superalgebra whose odd elements are $\tilde{Q}$ and $\hat{Q}$
while the even elements are $J^{1},J^{2},J^{3},J^{a}J^{a}$ and the unity
operator.

Alternatively, we can apply one more integral of motion

$\displaystyle Q=(\sigma^{a}L^{a}+1)P$ (106)

where $P$ is the parity operator. Operators (102) and (106) commute each
other, and so we can indicate four commuting integrals of motion, say,
$J^{3},\ J^{a}J^{a},\ \hat{Q}$ and $Q$ which make our system with three
spatial and one spin degree of freedom integrable. The remaining integrals of
motion guarantee the superintegrability of the considered system. For the
other examples of the integrals of motion for the SP equations including the
parity and some other discrete symmetries see papers [16] and [17].

Notice that up to the matrix multiplier $\gamma^{0}$ operator $\tilde{Q}$
coincides with the ”Dirac constant of motion” for the Dirac equation with
spherically symmetric potential of the external electric field. In equation
(106) matrix $\gamma^{0}$ is changed by the parity operator, and in this form
$Q$ is valid for the Dirac equation also provided $P$ is the Dirac parity
operator.

The considered models and the generic discussion presented above looks as a
good stimulation to study superintegrable systems which include both the Pauli
and spin-orbit couplings. Indeed, the correct QRSE should include both of
them. We remind that the superintegrability aspects of systems with spin-orbit
coupling (but without Pauli one) and Pauli coupling (but without spin-orbit
one) are relatively well studied, refer to [13, 14, 15] and [18, 19, 20]. Thus
the next step is to investigate the systems including both these couplings.

One more challenge is to classify symmetries of the QRSE with Hamiltonian
(60). In the case when the external electric field is trivial this equation is
reduced to the SP equation for neutral particles whose symmetries are
classified in recent paper [17].

Acknowledgement

The author is indebted to the National Academy of Science of the Ukraine for
the financial support in frames of the Programm of Support of the Priority
Research, project 7/1/241.

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