$0\le d<1$
:::info
$d\neq0$
:::
Possible conditions:
$v_0(v,t)=\left(v+{a\over d}\right)\cdot(1-d)^{-t}-{a\over d}$
$v_0(p,t)=\left(p+a\left({1+kd\over d}\right)t\right)\cdot{d\over1-(1-d)^t}-{a\over d}$
$v_0(v,p)=a\cdot\left({(1+kd)\over\ln(1-d)}W\left({\ln(1-d)\over a(1+kd)}\left(v+{a\over d}\right)(1-d)^\left({v+pd\over a(1+kd)}+{1\over d(1+kd)}\right)\right)-{1\over d}\right)$

First 3 relations:
$\left(v_1+{a\over d}\right)\cdot(1-d)^{-t_1}-{a\over d}=\left(v_2+{a\over d}\right)\cdot(1-d)^{-t_2}-{a\over d}$
${a\over d}\cdot\left((1-d)^{t_2}-(1-d)^{t_1}\right)=v_2(1-d)^{t_1}-v_1(1-d)^{t_2}$
$a(v_1,t_1,v_2,t_2)=d\cdot{v_2(1-d)^{t_1}-v_1(1-d)^{t_2}\over(1-d)^{t_2}-(1-d)^{t_1}}$

$\left(p_1+a\left({1+kd\over d}\right)t_1\right)\cdot{d\over1-(1-d)^{t_1}}-{a\over d}=\left(p_2+a\left({1+kd\over d}\right)t_2\right)\cdot{d\over1-(1-d)^{t_2}}-{a\over d}$
$a\left({1+kd\over d}\right)\left(t_2\left(1-(1-d)^{t_1}\right)-t_1\left(1-(1-d)^{t_2}\right)\right)=p_1\left(1-(1-d)^{t_2}\right)-p_2\left(1-(1-d)^{t_1}\right)$
$a(p_1,t_1,p_2,t_2)=\left(d\over1+kd\right){p_1\left(1-(1-d)^{t_2}\right)-p_2\left(1-(1-d)^{t_1}\right)\over t_2\left(1-(1-d)^{t_1}\right)-t_1\left(1-(1-d)^{t_2}\right)}$

$\left(v_1+{a\over d}\right)(1-d)^\left({v_1+p_1d\over a(1+kd)}\right)=\left(v_2+{a\over d}\right)(1-d)^\left({v_2+p_2d\over a(1+kd)}\right)$
$\left(v_1d+a\over v_2d+a\right)^a=(1-d)^\left({\Delta v+\Delta pd\over1+kd}\right)$
$f(x)=\left(v_1d+x\over v_2d+x\right)^x-(1-d)^\left({\Delta v+\Delta pd\over1+kd}\right)$

Last 3 relations:
$\left(v_1+{a\over d}\right)\cdot(1-d)^{-t_1}-{a\over d}=\left(p_2+a\left({1+kd\over d}\right)t_2\right)\cdot{d\over1-(1-d)^{t_2}}-{a\over d}$
$a\cdot\left({1-(1-d)^{t_2}\over d^2(1-d)^{t_1}}-\left(1+kd\over d\right)t_2\right)=p_2-v_1\cdot{1-(1-d)^{t_2}\over d(1-d)^{t_1}}$
$a(v_1,t_1,p_2,t_2)={p_2d(1-d)^{t_1}-v_1\cdot\left(1-(1-d)^{t_2}\right)\over{1-(1-d)^{t_2}\over d}-t_2(1+kd)\cdot(1-d)^{t_1}}$

${\ln(1-d)\over a(1+kd)}\left(v_1+{a\over d}\right)\cdot(1-d)^{-t_1}=W\left({\ln(1-d)\over a(1+kd)}\left(v_2+{a\over d}\right)(1-d)^\left({v_2+p_2d\over a(1+kd)}+{1\over d(1+kd)}\right)\right)$
${\ln(1-d)\over a(1+kd)}\left(v_1+{a\over d}\right)\cdot(1-d)^\left({1\over1+kd}\left({v_1\over a}+{1\over d}\right)\cdot(1-d)^{-t_1}-t_1\right)={\ln(1-d)\over a(1+kd)}\left(v_2+{a\over d}\right)(1-d)^\left({v_2+p_2d\over a(1+kd)}+{1\over d(1+kd)}\right)$
$\left(v_1+{a\over d}\right)\cdot(1-d)^\left({(1-d)^{-t_1}-1\over d(1+kd)}-t_1\right)=\left(v_2+{a\over d}\right)(1-d)^\left({v_2-v1(1-d)^{-t_1}+p_2d\over a(1+kd)}\right)$
$\gamma=(1-d)^\left({(1-d)^{-t_1}-1\over d(1+kd)}-t_1\right)$
$\left(\gamma{v_1d+a\over v_2d+a}\right)^a=(1-d)^\left({v_2-v_1(1-d)^{-t_1}+p_2d\over1+kd}\right)$
$g(x)=\left(\gamma{v_1d+x\over v_2d+x}\right)^x-(1-d)^\left({v_2-v_1(1-d)^{-t_1}+p_2d\over1+kd}\right)$

${\ln(1-d)\over(1+kd)}\left({p_1\over a}+\left({1+kd\over d}\right)t_1\right)\cdot{d\over1-(1-d)^{t_1}}=W\left({\ln(1-d)\over a(1+kd)}\left(v_2+{a\over d}\right)(1-d)^\left({v_2+p_2d\over a(1+kd)}+{1\over d(1+kd)}\right)\right)$
$\left(p_1+a\left({1+kd\over d}\right)t_1\right)\cdot{d\over1-(1-d)^{t_1}}(1-d)^\left({1\over(1+kd)}\left({p_1\over a}+\left({1+kd\over d}\right)t_1\right)\cdot{d\over1-(1-d)^{t_1}}\right)=\left(v_2+{a\over d}\right)(1-d)^\left({v_2+p_2d\over a(1+kd)}+{1\over d(1+kd)}\right)$
$\left(p_1d+a(1+kd)t_1\over{v_2d+a}\right)\cdot{d\over1-(1-d)^{t_1}}(1-d)^\left({1\over(1+kd)}\left({p_1\over a}+\left({1+kd\over d}\right)t_1\right)\cdot{d\over1-(1-d)^{t_1}}\right)=(1-d)^\left({v_2+p_2d\over a(1+kd)}+{1\over d(1+kd)}\right)$
$\left(p_1d+a(1+kd)t_1\over{v_2d+a}\right)\cdot{d\over1-(1-d)^{t_1}}(1-d)^\left({t_1\over1-(1-d)^{t_1}}-{1\over d(1+kd)}\right)=(1-d)^\left({v_2+d\cdot\left(p_2-{p_1\over1-(1-d)^{t_1}}\right)\over a(1+kd)}\right)$
$\gamma={d\over1-(1-d)^{t_1}}(1-d)^\left({t_1\over1-(1-d)^{t_1}}-{1\over d(1+kd)}\right)$
$\left(\gamma{p_1d+a(1+kd)t_1\over{v_2d+a}}\right)^a=(1-d)^\left({v_2+d\cdot\left(p_2-{p_1\over1-(1-d)^{t_1}}\right)\over1+kd}\right)$
$h(x)=\left(\gamma{p_1d+x(1+kd)t_1\over{v_2d+x}}\right)^x-(1-d)^\left({v_2+d\cdot\left(p_2-{p_1\over1-(1-d)^{t_1}}\right)\over1+kd}\right)$

$f(x),g(x),h(x)\in\left\{\left(ax+b\over x+c\right)^x-k\right\}$
$\delta(x)=\left(ax+b\over x+c\right)^x=z^\left(b-cz\over z-a\right)\qquad r(x)={b-cx\over x-a}$
$\displaystyle\lim_{x\to0^+}r(x)=-{b\over a}$
$\displaystyle\lim_{x\to a^+}r(x)=\begin{equation}\begin{cases}+\infty\quad b-ac>0\\-\infty\quad b-ac<0\end{cases}\end{equation}$
$\displaystyle\lim_{x\to a^-}r(x)=\begin{equation}\begin{cases}-\infty\quad b-ac>0\\+\infty\quad b-ac<0\end{cases}\end{equation}$
$\displaystyle\lim_{x\to+\infty}r(x)=-c$
$\delta'(z)=\left(\ln(z){ac-b\over(z-a)^2}+{b-cz\over z(z-a)}\right)\cdot\delta(z)$
$\delta'(z)>0\implies(ac-b)z\ln z>(cz-b)(z-a)\qquad z\neq a$
$\phi(x)=(ac-b)x\ln x-(cx-b)(x-a)\qquad x\neq a$
$\displaystyle\lim_{x\to0^+}\phi(x)=-ab\qquad\displaystyle\lim_{x\to+\infty}\phi(x)=\begin{equation}\begin{cases}+\infty\quad c<0\lor(c=0\land b<0)\\-\infty\quad c>0\lor(c=0\land b>0)\end{cases}\end{equation}$
$\phi'(x)=(ac-b)(1+\ln x)-2cx+ac+b$
$\phi'(x)=0\implies{2c\over b-ac}xe^\left(2cx\over b-ac\right)={2c\over b-ac}e^\left(2ac\over b-ac\right)\implies x={b-ac\over 2c}W\left({2c\over b-ac}e^\left(2ac\over b-ac\right)\right)$
:::info
$d=0$
:::
Possible conditions:
$v_0(v,t)=v+at$
$v_0(p,t)={p\over t}+{a\over2}(t-1+2k)$
$v_0(v,p)=ka-{a\over2}\pm\sqrt{\left(ka-{a\over2}-v\right)^2+2ap}$

First 3 relations:
$v_1+at_1=v_2+at_2$
$a(v_1,t_1,v_2,t_2)=-{\Delta v\over\Delta t}$

${p_1\over t_1}+{a\over2}(t_1-1+2k)={p_2\over t_2}+{a\over2}(t_2-1+2k)$
$a(p_1,t_1,p_2,t_2)={2\over\Delta t}\left({p_1\over t_1}-{p_2\over t_2}\right)$

$\left(ka-{a\over2}-v_1\right)^2+2ap_1=\left(ka-{a\over2}-v_2\right)^2+2ap_2$
$k^2a^2+{a^2\over4}+v_1^2-ka^2-2kv_1a+v_1a+2ap_1=k^2a^2+{a^2\over4}+v_2^2-ka^2-2kv_2a+v_2a+2ap_2$
$\left(k^2+{1\over4}-k\right)a^2+\left(2p_1+(1-2k)v_1\right)a+v_1^2=\left(k^2+{1\over4}-k\right)a^2+\left(2p_2+(1-2k)v_2\right)a+v_2^2$
$a(v_1,p_1,v_2,p_2)={v_1^2-v_2^2\over2\Delta p+(1-2k)\Delta v}$

Last 3 relations:
$v_1+at_1={p_2\over t_2}+{a\over2}(t_2-1+2k)$
$a(v_1,t_1,p_2,t_2)=2{v_1t_2-p_2\over t_2(t_2-2t_1-1+2k)}$

$v_1+at_1=ka-{a\over2}\pm\sqrt{\left(ka-{a\over2}-v_2\right)^2+2ap_2}$
$\left(ka-{a\over2}-v_1-at_1\right)^2=\left(ka-{a\over2}-v_2\right)^2+2ap_2$
$v_1^2+a^2t_1^2+v_1a(1-2k)+a^2t_1(1-2k)+2av_1t_1=v_2^2+v_2a(1-2k)+2ap_2$
$t_1(2k-1-t_1)a^2+\left(\Delta v(1-2k)+2(p_2-v_1t_1)\right)a+(v_2^2-v_1^2)=0$

${p_1\over t_1}+{a\over2}(t_1-1+2k)=ka-{a\over2}\pm\sqrt{\left(ka-{a\over2}-v_2\right)^2+2ap_2}$
$\left({p_1\over t_1}+{a\over2}t_1\right)^2=\left(ka-{a\over2}-v_2\right)^2+2ap_2$
${p_1^2\over t_1^2}+{a^2\over4}t_1^2+ap_1=k^2a^2+{a^2\over4}+v_2^2-ka^2-2kav_2+av_2+2ap_2$
$\left(k^2-k+{1-t_1^2\over4}\right)a^2+\left(v_2(1-2k)+2p_2-p_1\right)a+\left(v_2^2-{p_1^2\over t_1^2}\right)=0$