_{z}= e

^{zLE}

_{E}denotes Lie derivative along the vector field $E$. To any smooth function $J\; \in \; C\infty (M)$ this group assigns orbit that goes through $J$

_{z}(J) = e

^{zLE}(J) = J + zL

_{E}J + ½z

^{2}(L

_{E})

^{2}J + ⋯

_{E}J) = c[E , W](dJ) c ∈ ℝ∖(0∪ℕ)

_{E}(J) is involutive.

_{E}J) + W(d(L

_{E})

^{2}J) = c[E,[E , W]](dJ) + c[E , W](dL

_{E}J)

_{E})

^{2}J) = (c − 1)[E , W](dL

_{E}J)

_{E})

^{m + 1}J) = (c − m)[E , W](d(L

_{E})

^{m}J)

_{E})

^{m}J are in involution. Indeed

^{(k)}, J

^{(m)}} = W(dJ

^{(k)}∧ dJ

^{(m)})

^{(k)}∧ dJ

^{(m)}) = W(d(L

_{E})

^{k}J ∧ dJ

^{(m)}) = L

_{W(d(LE)kJ)}J

^{(m)}

_{[E , W](d(LE)k − 1J)}J

^{(m)}= (c − k + 1)[E , W](dJ

^{(k − 1)}∧ dJ

^{(m)})

_{[E , W](d(LE)mJ)}J

^{(k − 1)}= −

_{W(d(LE)m + 1J)}J

^{(k − 1)}

^{(k − 1)}∧ dJ

^{(m + 1)})

^{(k)}, J

^{(m)}} = (c − k + 1){J

^{(k − 1)}, J

^{(m + 1)}}

^{(k)}, J

^{(m)}} = {J

^{(m)}, J

^{(k)}}

^{(k)}, J

^{(m)}} = 0

_{E})

^{m}J are in involution. In the same time orbit $J(z)$ is linear combination of functions $J(m)$ and thus it is involutive as well.

^{(m + 1)})

^{(m)}, f}

^{(m)})

^{(m + 1)}), W(dJ

^{(m)})]

^{(m + 1)}) = (c − m)[E , W](dJ

^{(m)}) + (c − m)W(dL

_{E}J

^{(m)})

^{(m + 1)},J

^{(m)}}) − (c − m + 1)W(dJ

^{(m + 1)})

^{(m + 1)}) + (c − m)W(dJ

^{(m + 1)}) − (c − m + 1)W(dJ

^{(m + 1)}) = 0

_{E})

^{m}J form Lenard scheme with respect to bi-Hamiltonian structure formed by Poisson bivector fields $W$ and $[E\; ,\; W]$ (see [1],[4]).

_{E}J) = − [E , W](dJ)

_{t}= cv

_{xx}+ u

_{x}v + uv

_{x}

_{t}= − cu

_{xx}+ uu

_{x}+ 3vv

_{x}

^{2}v + v

^{3}+ 2cuv

_{x})dx

_{x}+ B ∧ B

_{x})dx

_{x}− cvA

_{xx}+ (uu

_{x}+ vv

_{x})B + (u

^{2}+ 2v

^{2})B

_{x}+ cuB

_{xx})xdx

_{x}+ x((uv)

_{x}+ cv

_{xx}))A

^{2}+ 2v

^{2}− 2cu

_{x}+ x(uu

_{x}+ 3vv

_{x}− cu

_{xx}))B]dx

^{(0)}=

^{(1)}= L

_{E}J

^{(0)}=

^{2}+ v

^{2})dx

^{(2)}= (L

_{E})

^{2}J

^{(0)}= m

^{2}v + v

^{3}+ 2cuv

_{x})dx

^{(3)}= (L

_{E})

^{3}J

^{(0)}=

^{4}+ 5v

^{4}+ 6u

^{2}v

^{2}

^{2}u

_{x}+ 4c

^{2}u

_{x}

^{2}+ 4c

^{2}v

_{x}

^{2})dx

^{(m)}= (L

_{E})

^{m}J

^{(0)}= L

_{E}J

^{(m − 1)}

_{t}= cu

_{xx}+ 2uu

_{x}

_{t}= − cv

_{xx}+ 2uv

_{x}+ 2u

_{x}v

^{2}v + cu

_{x}v)dx

_{x}dx

^{2}A

_{x}− cuA

_{xx}+ (uv)

_{x}B + 3uvB

_{x}+ cvB

_{xx})xdx

^{2}+ 2cu

_{x}+ x(2uu

_{x}+ cu

_{xx}))A

_{x}+ x(2(uv)

_{x}− cv

_{xx}))B]dx

^{(0)}=

^{(1)}= L

_{E}J

^{(0)}= m

^{(2)}= (L

_{E})

^{2}J

^{(0)}= 2m

^{2}v + cu

_{x}v)dx

^{(3)}= (L

_{E})

^{3}J

^{(0)}= 3m

^{3}v − 3cu

^{2}v

_{x}− 2c

^{2}u

_{x}v

_{x})dx

^{(m)}= (L

_{E})

^{m}J

^{(0)}= L

_{E}J

^{(m − 1)}

_{t}= − 2FGU

_{xx}+ 〈U , GU

_{x}〉C + 〈C , GU

_{x}〉U + 〈C , GU〉U

_{x}

^{T}= G, F

^{T}= − F

_{mn}C

_{k}+ F

_{km}C

_{n}+ F

_{nk}C

_{m}= 0

_{m}that are smooth functions on $\mathbb{R}2$ subjected to zero boundary conditions

_{m}= u

_{m}(x, t); u

_{m}(±∞, t) = 0; m = 1 ... D

_{mn}C

_{k}+ F

_{km}C

_{n}+ F

_{nk}C

_{m}= 0

_{m}Y

_{m}.

^{−1}A

_{x}〉dx

_{m}that are vector fields defined for every smooth functional $R(u)$ via variational derivatives $A$

_{m}(R) = δR/δu

_{m}. Moreover this model is actually bi-Hamiltonian as there exist another invariant Poisson bivector

_{x}〉 + 〈A

_{x}, FA

_{x}〉}dx

_{x}, GU〉}dx

_{x}〉 + 〈U , GU〉〈C , A

_{x}〉

_{x}〉〈C , A〉 + 2〈FGU , A

_{xx}〉}xdx

_{x}, A〉

_{x}〉〈U , A〉 + 〈C , GU〉〈U

_{x}, A〉

_{x}〉〈C , A〉 + 2〈FGU

_{xx}, A〉)}dx

^{(0)}=

^{(1)}= L

_{E}J

^{(0)}= ½〈C , K〉

^{(2)}= (L

_{E})

^{2}J

^{(0)}= 〈C , K〉

_{x}, GU〉}dx

^{(3)}= (L

_{E})

^{3}J

^{(0)}= ¼〈C , K〉

^{2}

^{2}〈U , GU〉 + 32〈C , GU〉〈GU , FGU

_{x}〉

_{x}〉 + 48〈FGU

_{x}, GFGU

_{x}〉}dx

^{(m)}= (L

_{E})

^{m}J

^{(0)}= L

_{E}J

^{(m − 1)}

_{12}= − F

_{21}= ½c, C = K = (0 , 1), G = 1

_{12}= − F

_{21}= ½c, C = K = (0 , 1)

_{12}= G

_{21}= 1, G

_{11}= G

_{22}= 0